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AN ELEMENTARY TREATISE 
ON 

CROSS-RATIO GEOMETRY 



CAMBRIDGE UNIVERSITY PRESS 

ILontJon: FETTER LANE, E.G. 

C. F. CLAY, Manager 




ffiBinburgI): loo, PRINCES STREET 

ISctlin: A. ASHER AND CO. 

ILeipJig: F. A. BRGCKHAUS 

ilcfaJ lork: G. P. PUTNAM'S SONS 

aSombag anU GTalcutta: MACMILLAN AND CO., Ltd. 



All rights reserved 



AN ELEMENTARY TREATISE 




ON 



CROSS-RATIO GEOMETRY 



WITH HISTORICAL NOTES 



BY THE 

REV. JOHN J. MILNE, M.A 

ST John's college, Cambridge. 

AUTHOR OF WEEKLY PROBLEM PAPERS, &c. 



DEPARTMEI^iT OF MATHUMATIGS 
UNIVERSITY OF TORpNT© 



Cambridge : 

at the University Press 

191 1 






CambriUgc : 



PRINTED BY JOHN CLAY, M.A. 
AT THE UNIVERSITY PRESS 



1Sg5 



PREFACE 

THE development of the theory of cross-ratio is due, quite 
independently of each other ^, to Mobius, Der harycentrische 
Calcul, 1827, and to Chasles, Aper^u Historique, 1829 — 1837, 
followed by the Geometrie Superieure, 1852, where the subject is 
treated very fully as regards the point and straight line, its 
application to the conic being given in the Traite des Sections 
Coniques, 1865. Some employment of its principles is met with 
in the various treatises on what is sometimes called Modern 
Geometry which have subsequently appeared, but as far as I am 
aware there is no English text-book exclusively devoted to it. 

The power of the method of cross- ratio, as an instrument of 
analysis, it is not easy to over-rate. In the facility with which it 
deals alike with the range and pencil, with the points and line at 
infinity, with questions relating to concurrency and coUinearity, 
loci and envelopes, it can compare not unfavourably with the 
methods of analytical geometry, and in those questions to which 
it is specially applicable, the steps necessary to establish any 
result are few in number, and are mostly of the same character, 
dealing as a rule with the homography of certain ranges or pencils, 
with the additional advantage that the geometrical meaning of 
each step is in general obvious. 

1 See the note on p. xxxii of the Preface to Chasles' Geometrie Superieure, 
where in speaking of the Calcul harycentrische he says "ce que je n'ai su que 
fort longtemps apres la pubUcation de I'Aper^u historique." 



VI PREFACE 

Again, in dealing with pairs of imaginary points, analytical 
geometry is generally content with the recognition of their occur- 
rence owing to certain relations between the coefficients of an 
equation ; but the theory of cross-ratio goes further, and not only 
gives us the geometrical conditions under which they occur, but 
it gives us the actual position of their mid-point, and the value of 
the rectangle formed by the segments joining them to a real point. 

This treatise naturally divides itself into two parts. In 
Chapters I — X, which deal exclusively with the point and 
straight line, the only knowledge of geometry which the reader 
is assumed to possess is that of the fundamental properties of 
similar triangles and ratio, and I have thought it advisable to 
make this part of the subject quite self-contained. It is usual to 
discuss co-axial ranges by projecting them on to a conic or circle, 
and making use of the Pascal line, &c., but by means of Prof. A. 
Lodge's method, given in Chap. VII, a student is enabled to 
construct two co-axial ranges, and to find their common points, 
(fee, without interrupting the logical course of his reading. . 

In dealing with involution it seemed most simple and natural 
to treat it as the case of two co-axial homographic rows in which 
/ and J', the correspondents of points at infinity, coincide. 

In the second part, beginning with Chap. XI, I have adopted 
B. W. Home's method of applying the theory of cross-ratio to the 
conic, which obviates the necessity of first proving properties for 
the circle, and then by projection obtaining the corresponding 
properties for the conic. This requires the knowledge on the 
part of the student of four elementary propositions in geometrical 
conies, viz. those given in Arts. 127, 128, 135, 136, and I have 
had no hesitation in assuming them for two reasons. In the first 
place, this part of the work is intended to be a treatise, not on 
geometrical conies, but on the application of the theory of cross- 
ratio to the subject; and secondly, although the subject of 
geometrical conies can be developed by means of the theory of 
cross-ratio, as Chasles has shewn in his fascinating Traite des 




PREFACE Vll 

Sections Coniques, I am strongly of opinion that a student 
ought to have obtained from one of the ordinary text-books a 
working knowledge of its elements before he is introduced to the 
theorems of Pascal, Brianchon, Desargues, &c., which take him 
at once to more advanced work. Another reason is that by the 
aid of the theory of cross-ratio it is just as easy to prove properties 
of conies, considered separately or as a system, as it is to prove the 
corresponding properties for the circle, in fact in some cases it is 
easier and more complete, as we might expect from the con- 
sideration that the circle is only a particular case of the conic. 

As there is scarcely any part of conies to which the theory of 
cross-ratio is not applicable, and as I wished to curtail the size of 
the book as much as possible, it was necessary to follow some 
definite path, and I have selected the course which leads us to 
consider conies through four points, and conies touching four 
lines, their common chords and tangents, the relations between 
the four constants of homology obtained by taking any pair of 
common chords with the pair of corresponding tangent vertices as 
axes and centres of homology, and conies having double contact. 
I took this route because it contains parts of the subject which 
have not previously been fully treated, and at the same time it 
gives the student a good illustration of the power of the theory. 
A good deal of the work in these chapters is original, and where 
it is not so, references have been given, where possible, to the 
original authorities. 

I have thought it advisable to give a figure with almost every 
proposition so that the student may be enabled readily to follow 
all that he reads, and to remove any feeling of indefiniteness 
in his ujind I have given full solutions in the case of problems 
which depend on finding the common points of two co-axial 
ranges. With the same object I have given complete figures 
in the different cases of the real and ideal common chords of two 
conies, and their common self- conjugate triangle. 

The reader will notice that throughout the work I have made 



Vlll PREFACE 

no use either of projection, except in Arts. 138, 139, and in 
Chap. XIX (which deals with generalised projection), or of the 
principle of duality. My reason for the omission of the first is 
that it was not necessary for my purpose, and with regard to the 
latter the direct demonstration of a correlative theorem gave me 
an additional opportunity of illustrating the use of the theory of 
cross-ratio. 

I have given at intervals throughout the work historical 
notes illustrative of the subject as far as it was in my power to 
do so within the limits of a private library, and by means of 
books kindly lent from the library of St John's College, Cam- 
bridge, and in doing this, one of my objects has been to shew 
that both parts of the subject are based upon ancient geometry, 
the theorem that a pencil cuts all transversals in equicross ratios 
being given by Pappus, and the converse of the anharmonic 
property of conies being due to Apollonius. With the same 
purpose in viev/ I have given in an Appendix Pappus' account of 
the lost books of Euclid's Porisms, so that the student may have 
the opportunity of forming an opinion as to the probability of 
their connection with the theory of cross-ratio. 

As the term *' Modern Geometry" is frequently used without 
it being stated whether the adjective refers to the matter or the 
methods employed, or both, the following brief statement respect- 
ing the text-books on geometry in common use by the ancients 
will give the reader a general idea of the amount of knowledge of 
the subject which they possessed. 

Pappus, in the preface to the seventh book of his Mathematical 
Collections, tells us that when a student had read the Elements 
of Euclid, and wished to proceed to more advanced work, the 
following was the order of the books which he would take up. 

I. Euclid's Data^ one book containing 100 theorems. 

This is still extant, and to be met with in some of the older 
editions, e.g. that by Barrow 1732, and by R. Simson 1841. 



PREFACE IX 

The following works by Apollonius : 

II. Proportional Section^ two books containing 181 theorems. 
This was discovered in an Arabic MS in the Bodleian 

Library, and a Latin translation was published by Halley in 
1706. See Art. 88. 

III. Spatial Section, two books containing 124 theorems. 
This has been " restored " by Halley, and published with the 

books on Proportional Section. Another restoration was made by 
Snell 1607. 

IV. Determinate Section, two books containing 83 theorems. 
This has been restored by various geometers. Snell 1601, 

Lawson 1772, Wales 1772, Simson 1776. 

V. Tangencies, two books, 81 theorems. 
Restored by Yieta 1600, Lawson 1771. 

VI. Euclid's Porisms, three books. See Appendix I. 
The following by Apollonius : 

VII. Inclinations, two books, 125 theorems. 

This was a treatise respecting lines which pass through a 
given point whilst satisfying certain conditions [e.g. through a 
given point to draw a straight line such that the part of it inter- 
cepted between two given straight lines may be of given length). 
Restored by Ghetaldus 1607, Horsley, 1770, Burrow 1779. 

VIII. Plane loci, two books, 147 theorems. 
Restored by Schooten 1656, Fermat 1679, Simson 1749. 

IX. Conies, eight books, 487 theorems. 

Books I — IV are extant in Greek, V — VII were discovered 
in Arabic and translated into Latin by Ecchellensis and Borellus 
in 1661. In 1710 Halley published the first four books in Greek 

65 



X PREFACE 

and Latin, and the next three in Latin, together with a conjectural 
restitution of the 8th book, which is still missing \ 

X. Aristaeus, solid loci, five books. 

XL Euclid, loci ad superficiem. 

XII. Eratosthenes, on Means, two books. 

Of these, all except I, II and IX are lost, although it is quite 
possible they may still be in existence, probably in Arabic. 

I have purposely refrained from giving a large number of 
Examples, and those given (260) have been carefully chosen to 
illustrate the text. The student who requires more will find 
admirable collections in Russell's Elementary Treatise on Pure 
Geometry (1905), and in Durell's Plane Geometry for Advanced 
Students, Part II (1910). 

I take this opportunity of acknowledging my personal obliga- 
tions to Prof. A. Lodge, of Charterhouse (late Professor of Pure 
Mathematics at Cooper's Hill), for the stimulating interest he has 
taken in the book throughout. He read through the whole of 
the work in manuscript, and again when it was passing through 
the press, and was of the greatest help in discussing the difficulties 
which arose from time to time; in fact he could not have taken a 
greater interest in it if the work had been his own, and it is 
chiefly owing to his friendly persistence that the treatise, which 
was originally written to gratify my own interest in the subject, 
has seen the light. 

I am also greatly indebted to Prof. Heawood, of Durham, who 
kindly read through the proof-sheets, and from whom I received 
many valuable criticisms and suggestions. 

1 For a fuller account^ see the Math. Gazette for October 1895. 

JOHN J. MILNE. 

Lee-on-the-Solent. 
September^ 1911. 



CONTENTS 

CHAPTER I 
Cross-ratio of a range of four points, and of a 

PENCIL OF four LINES 
ARTICLES 

1 Sense of lines and angles. 

2 — 4 Eelations between the cross-ratios of four points. 

5 Cross-ratios represented geometrically by three collinear points. 

9, 10 To find the fourth point of a range, given the value of the cross- 
ratio and the positions of three points. 
11 — 13 Cross-ratios represented geometrically and trigonometrically. 
14 — 16 Cross-ratio of a pencil of four rays is that of the range on any 
transversal. 

17 Construction of the fourth ray of a pencil given the value of the 

cross-ratio and the positions of three rays. 

18 Menelaus' Theorem. Historical note. 
Examples. 

CHAPTER II 

Equicross ranges and equicross pencils. Perspeciive 

19, 20 Equality of two ranges or pencils which are such that a cross- 
ratio of the one is equal to a cross-ratio of the other. 
21 Two ranges cut by the same pencil are equicross. Eanges in 

perspective. 
22, 23 Equicross ranges. 

24, 25 Two pencils subtended by the same range are equicross. Pencils 
in perspective. 
26 Co-axial and co-polar triangles, triangles in perspective. 



XU CONTENTS 

CHAPTER III 
Harmonic ratio 

ARTICLES 

27 Definition of harmonic ratio, harmonic conjugate points. 

28 Values of the cross-ratios of a harmonic range. A range is 

harmonic if the value of its cross-ratio is unaltered by inter- 
changing one pair of points separately. 

29 To find the fourth harmonic of three given points. If one of 

three points is midway between the other two, its conjugate is 
at infinity. 

30 To construct the fourth harmonic of three given rays. If one of 

three rays bisects the angle between the other two, its conjugate 
is at right angles to it. 
32 — 34 Relations between the segments of a harmonic range. 

35 On a given straight line to find a segment which will divide two 

given segments harmonically. 

36 Given two pairs of lines through the same point, to draw through 

it another pair of lines which will form a harmonic pencil with 
each of the given pairs. 



CHAPTER IV 

HOMOGRAPHIC RANGES AND HOMOGRAPHIC PENCILS 

37, 38 nomographic ranges. Characteristic of a range. 

39 Two ranges which are homographic to the same range are homo- 

graphic to one another. 

40 To construct a range on a line V homographic to a given range 

on a line L. 

41 A pencil cuts any two transversals homographically. Common 

points, ranges in perspective. 
42, 43 Homographic pencils, 

44 Two pencils which are homographic to the same pencil are homo- 

graphic to one another. 

45 Pencils in perspective are homographic. Common rays. 



CONTENTS Xlll 

ARTICLES 

46 To construct a range in perspective with two homographic 

ranges. 

48 Given three rays of a pencil, to find a fourth ray so that the 

pencil may be equicross with a given pencil. 

49 Classes of questions solved by properties of ranges or pencils in 

perspective. 
Examples illustrating the above principles. 

CHAPTER Y 

Cross-axis. Cross-centre 

50, 51 Construction and history of homographic or cross-axis. 

53 Employed to divide a line homographically to a given divided 

line. 

54 Given the cross-axis of two ranges, to find the point on each 

range corresponding to the point at infinity on the other. 

55 The line IJ' is parallel to the cross-axis. 

57 Cross-centre property of two homographic pencils. 

58 To find in each of two homographic pencils the ray corresponding 

to the ray joining the centres of the pencils. 
59, 60 The homographic or cross-centre. 

61 To construct a pencil homographic to a given pencil. 

CHAPTER VI 

Metrical properties of homographic ranges. The constant 
OF correspondence. Homographic equations. One-to- 
one correspondence 

62 Investigation of the equation - — = 11 t-t-, • 

om om 

63 Determination of the lengths al, a' J'. 

64 The product Im . J'm' is constant. 

65 The value of the power of the correspondence. 
66 — 68 Equations representing two homographic ranges. 
67, 68 Proportional Section. 

69 One-to-one correspondence. Factorising homography. 

Examples illustrating the meaning and application of homo- 
graphic equations. 



XIV CONTENTS 



CHAPTER VII 

Two HOMOGRAPHIC CO-AXIAL KANGES. ThEIR COMMON POINTS, 
AND THE METHODS OP FINDING THEM 

ARTICLES 

70 The divisions of homographic ranges on two lines are independent 

of the angle between the lines. 

71 Equation giving the positions of the common points e, f. 

72 The mid-point of ef is the mid-point of IJ'. 

73 Common points, when real and when imaginary. 
74 — 76 The cross-ratio {efvim') is constant. 

77 If two ranges are similar, one of the common points is at infinity. 

78 Conditions determining the reality of the common points. 

80 In general in two homographic pencils with different centres there 

are only two pairs of corresponding rays which intersect on a 
given transversal. 
Common rays of two homographic concentric pencils. 

81 Case where the common points are imaginary. Rotation centres. 

Rotation angles. 
82 — 87 Different methods of constructing two homographic co-axial 
ranges, and of finding their common points. 



CHAPTER VIII 

Problems of the three sections. Other problems whose 
solutions depend on finding the common points of two 
co-axial homographic ranges 

88 Historical account of the Problems of the three Sections- 

89 Determinate Section. 

90 Spatial Section. 

91 Proportional Section. 

92 To inscribe in a triangle ABC another triangle whose sides pass 

through three given points. Rule of false position. 
94, 95 Two other problems solved by the same methods. 
Examples. 



CONTENTS XV 

CHAPTER IX 
Involution 

ARTICLES 

96 An involution range consists of two homographic ranges in which 

the points I, J' coincide, and every point on the axis has its 
two corresponding points coincident. 

97 Conjugate points, double points, and centre of an involution 

range. 

98 Given three pairs of conjugate points, the range formed by four 

of the points is equicross with that formed by their conjugates. 

99 The range [aa'ef) is harmonic. 

102, 104 Given the characteristic of an involution range, to find its centre 
and the conjugate of any point. 
103 The value of the power of an involution. 

105 Eule to determine if six collinear points are in involution. 

106 Relations between the segments connecting six points in involu- 

tion. 

107 Case where the centre of an involution range is at infinity. 

108 The double points are real or imaginary according as the 

segments joining pairs of conjugate points do not, or do over- 
lap. 

110 One-to-one correspondence. 

111 Involution pencils. 

112 Orthogonal pencils. 

113, 114 Circular points at infinity. 
Examples. 

CHAPTER X 

Involution and harmonic section. Harmonic properties of 
a quadrangle and quadrilateral. pole and polar 

115 Relation between involution and harmonic section. 

116 An involution system can always be constructed, whether the 

double points are real or imaginary, 

117 Quadrilateral and quadrangle. 

118 Harmonic properties of a quadrilateral and quadrangle. 

To construct the fourth ray of a harmonic pencil when three of 
them are given. 



XVI CONTENTS 

ARTICLES 

119 A quadrilateral cuts any transversal in involution. 

120 The pencil formed by joining any point to the vertices of a quad- 

rangle is in involution. 

121 Pole and polar for an angle or pair of lines. 

122 — 125 Applications of pole and polar to a triangle and quadrangle. 
Examples. 

APPENDIX I 

Pappus' account of the Porisms of Euclid, and his 
Lemmas (I — XIX) on them 

Translation of Pappus' account of the three books of Euclid's 
Porisms, and his statement of the 29 classes in which they 
were arranged. 

Historical account of modern investigations into the subject. 

Translation of the enunciations of Pappus' 19 Lemmas on 
Euclid's Porisms, with modern proofs by the theory of cross- 
ratio. 

CHAPTER XI 

Anharmonic properties of points and tangents of a conic. 
The locus ad tres et quatuor lineas 

127, 128 Two elementary properties of conies. 

129 The cross-ratio of any conic-pencil formed from four fixed points 

on a conic is constant. 

130 The cross-ratio of the range on any tangent made by four fixed 

tangents is constant. 

131 The cross-ratio of a conic-pencil is equal to that of the correspond- 

ing tangent range. 

133 Partial converse of Art. 129. 

134 Partial converse of Art. 130. 

135, 136 Proofs of properties of Arts. 129, 130 based upon propositions 
given by ApoUonius. 

137 Anharmonic property of (1) points, (2) tangents of a conic. 

138 Intersections of corresponding rays of two homographic pencils 

lie on a conic. 



CONTENTS 



XVll 



139 Lines joining pairs of corresponding points of two homographic 

ranges envelop a conic. 

140 A conic can be drawn through five points. 

142 A conic can be drawn touching five straight lines. 
143, 144 Consideration of species of conies given by anharmonic properties 
of points and tangents. 
145 The locus ad tres et quatuor lineas, with a historical account. 
Values of the constants k-^ and /fg in ^^^ equations ap = Kiy^ and 
a^ = K^-y5. Connection between the locus and the anharmonic 
property of the points of a conic. 
Examples. 

CHAPTER XII 
Pascal — Brianchon — Newton — Maclaurin 

Theorems of Pascal and Brianchon. 

Newton's method of describing a conic. 

Maclaurin's Theorem and its correlative. 

Cross-axis and common points of two homographic ranges on a 

conic. 
To construct a range homographic to a given range on a conic, 

and to find their common points. 
If (a) and {a') are two homographic divisions on a conic, e, / 

their common points, {aa'ef) is constant, and conversely. 
160 By means of a conic to construct a range homographic to a 

co-axial range, and to find their common points. See also 

Arts. 83—86. 



146—149 

150 
151—154 

155 

156, 157 
159 



CHAPTER XIII 
Pole and polar. Conjugate points and lines. Circular 

POINTS AT infinity. DeSARGUES' ThEOREM AND ITS CORRE- 
LATIVE. Propositions respecting triangles, quadrangles 

AND quadrilaterals INSCRIBED IN AND CIRCUMSCRIBED 
ABOUT A CONIC. CONTRA-POLAR CONICS 
161, 162 Pole and polar for a conic. 

163 Intersection of two chords is the pole of the line joining their 
poles. 



XVlll CONTENTS 

ARTICLES 

164 Transversals through a fixed point determine a conic-pencil iu 

involution. Construction of a conic-pencil in involution. 
165 — 178 Conjugate points and lines. 

165 Self-conjugate triangle. 

166 Consideration of the meaning of the word ' conjugate.' 

167 Kange of poles on a straight line is homographic with the pencil 

formed by their polars. 

169 Consideration of the cases where the line joining two conjugate 

points meets the conic in (1) real, (2) imaginary points. 

170 Construction of a triangle self- conjugate for a conic. 

171 If PQ, P'Q' are conjugate chords, the conic-pencil {QP'PQ') is 

harmonic. 
172 — 176 Properties relating to conjugate points and lines. 

177 Pairs of tangents from points on a straight line determine an 

involution range on any tangent. 

178 A variable tangent is divided harmonically by twa fixed tangents, 

their chord of contact and the curve. 
A chord is cut harmonically by any tangent and the line joining 
its point of contact to the pole of the chord. 
179 — 183 Circular points at infinity, i, i'. 

179 Every circle passes through i, %'. 

If G is the centre of a circle, Ot, Ci' are asymptotes. 

180 Any conic through i, i' is a circle. 

181 If the pencil V {aa'ii') is constant, the angle aVa' is constant. 

182 If <S is a focus of a conic, Si, Si' are tangents. 

183 i,\i' are conjugate points for a rectangular hyperbola, and con- 

versely. 

184 Fregier's Theorem. 

185 The locus of the intersection of pairs of conjugate lines passing 

through two given points is a conic. 
A transversal through a fixed point meets a fixed line in d, and a 

conic in Q, R. If (PdQR) is harmonic, the locus of P is a 

conic. 
The locus of the intersection of tangents to a conic which divide 

a given segment of a line harmonically is a conic. 
The envelope of a chord of a conic divided harmonically by two 

given lines is a conic. 
The director circle. 
See also Art. 191 and Chap. XIX, Exs. 22—24. 



CONTENTS 



XIX 



ARTICLES 
186 

187, 188 
190 

191 



192 



193—195 
196—198 



199 
200 

201 



202—207 
202 



203 
204 



205 

207 
208 



Two homographic divisions on a conic. 

Desargues' Theorem and its correlative. 

Properties of a quadrangle inscribed in a conic, and of the quad- 
rilateral formed by drawing tangents at its vertices. 

Given a fixed point p, a fixed line L, a conic C, and any trans- 
versal through p meeting L in m and C in a, a! . P is the pole 
of L, and fx the fourth harmonic of m for a, a'. As the trans- 
versal rotates about p, the locus of /a is a conic, and the envelope 
of its polar is another conic. 

If (a6c ...), {a'b'c' ...) are two homographic divisions on a conic, 
e, f their common points, T the pole of ef, T {aa'ef) is constant, 
and conversely. 

Triangles inscribed in one conic and circumscribing another. 

If two triangles are self-conjugate to a conic, their six vertices lie 
on one conic, and their six sides touch another, &c. 

If a quadrilateral circumscribes one conic, its points of contact 
and two of its opposite vertices lie on another. 

If a quadrangle is inscribed in one conic, the tangents at 
its vertices and a pair of opposite sides are tangents to 
another, &c. 

a6, cd are two chords, P, Q their poles. If the conic-pencil 
(abed) = \, the pencil P (abed) = Q {abed) = \2. 

Cor. If {abc ...), {a'b'c'...) are two homographic divisions on a 
conic, e, f their common points, {aa'ef) is constant. 

Contra-polar or harmotomic conies and their chief properties. 

Two conies a, /3 intersect in A, B, I, I', and the poles of one of 
the chords as ir for a and /3 are T, C respectively. 

Then if TA is a tangent to /3, 

(1) TB will be a tangent to /3, 

(2) CA and CB will be tangents to a. 

To construct a conic contra-polar to a given conic. 

In two contra-polar conies the tangents at any one of the points 
of intersection, as A, divide the opposite chord IF harmonic- 
ally. 

In two contra-polar conies any transversal through either of the 
poles is divided harmonically by the two conies. 

The relations between contra-polar conies established by Art. 201. 

Important properties in connection with conjugate points and 
lines. 

Examples. 



XX CONTENTS 



CHAPTER XIV 

Common chords and common tangents op two or more conics 
(1) passing through four points, (2) touching four 
straight lines, i.e. of pencils and ranges of conics 

ARTICLES 

209 Two conies circumscribing a quadrangle. 

210 The polars of any point on a common chord intersect on the 

chord. 

211 The intersections of pairs of common chords form a common 

self- conjugate triangle. 

212 Two conics have only one common self-conjugate triangle. 
213, 214 Any transversal is cut by the system in involution. 

215 If three conics have a common chord, their three other corre- 

sponding common chords are concurrent. 

216 A pencil of conics cuts a transversal in involution. 

217 Two given points are conjugate for only one conic of a pencil. 

218 The polars of any point P all pass through another point P'. 

If two points are conjugate for two conics A , B they are conjugate 
for the pencil to which A , B belong. 
219 — 221 If P moves along a fixed line, P' describes a conic passing 
through eleven known points. 

222 The locus of the centres of the system is a conic through eleven 

points. 

223 Two conics inscribed in a quadrilateral. Tangent vertices. 

224 A pair of lines through a tangent vertex conjugate for one conic 

are also conjugate for the other. 

225 The line joining a pair of tangent vertices has the same pole. 

226 The lines joining three pairs of tangent vertices form a common 

self-conjugate triangle. 

227 The poles of a common chord lie on the line joining the corre- 

sponding tangent vertices. 
228—230 If from any point P two pairs of tangents PQ, PR ; PQ', PR' are 
drawn, the pencil P{QR, Q'R', T^T^ .,.) and the range 

{aa', bb', cc', T-^T^...) 
are in involution. 
231 If three conics have a common tangent vertex, their three other 
corresponding tangent vertices are collinear. 



CONTENTS XXI 

ARTICLES 

232 A system of conies inscribed in a quadrilateral. Eange of conies. 
A given pair of lines are conjugate for only one conic of a range. 
A pair of lines which are conjugate for two conies A, B are con- 
jugate for the range to which A, B belong. 

233 The poles of any straight line L lie on a straight line L'. 

234 — 237 If L rotates about a fixed point, L' envelops a conic touching 
eleven known lines. 

238 The locus of the centres of the range is a straight line. 

239 Let PQRS be a quadrangle, A, B, G its diagonal points, and X, Y 

two conjugate points for the pencil of conies circumscribing the 
quadrangle. Then X (PQRS) = Y{XABC). Dr W. P. Milne. 

240 Let PQRS be a quadrilateral, ABC its diagonal triangle, Ox, Oij 

two lines conjugate for the range of conies inscribed in the 
quadrilateral. Let one of them, Ox, meet the lines PQ, QR, 
RS, SP in the points p, q, r, s, and let the other, Oy, meet 
BC, GA, AB in a, ]8, y. Then {3^qrs) = {0a^'y). Correlative of 
Art. 239. 
Examples. 

CHAPTER XV 

The homologue op a line and conic. Relations between 
a pair of common chords and the corresponding pair 
of tangent vertices. relations between the four 
constants of homology 

241 Definitions. 

242 The homologue of a straight line is a straight line. 

243 The homologue of a conic is a conic. 

245 If from any point on a common chord we draw tangents to the 

conies, the lines joining the points of contact on the first conic 
to those on the second pass by pairs through the corresponding 
tangent vertices. 
247 If a transversal is drawn through a tangent vertex, the tangents 
at the points where it cuts the conies meet by pairs on the 
corresponding common chords. 
249 — 251 Kelations between the four constants of homology. Harmonic 
homology. 
Examples. 



XXll CONTENTS 

CHAPTER XVI 

Construction of the common chords, tangent vertices, and 

COMMON self-conjugate TRIANGLE OP TWO CONICS 

ARTICLES 

253 Common chords and tangent vertices are real, ideal or imaginary. 

Construction of the common chords, &c. 

254 When the conies intersect in four real separate points. 

255 When two of the points coincide. 

256 When the conies have double contact. 

257 When the conies osculate. 

258 When the conies have four consecutive points common. 

259 When the conies intersect in only two real points. 

260 When the conies touch externally. 

261 When the conies touch internally. 

262 When each conic is entirely without the other. 

263 When one conic is entirely within the other. 

CHAPTER XYII 

CONICS HAVING DOUBLE CONTACT 

265 Any pair of conjugate points on the chord of contact form with 
the pole of contact a common self-conjugate triangle. 

267 The polars of any point intersect on the chord of contact, and of 
two conjugate points, one is always on the chord of contact. 

269 Of two conjugate lines one passes through the pole of contact. 

272 Two conies having double contact with a third. 

273 If Tahb'a' is any transversal through the pole of contact T, the 

cross-ratio {Tbaa') is constant, and is the reciprocal of {Th'aa'). 
274: Chords joining pairs of corresponding points of two homographic 
rows on a conic envelop a second conic having double contact 
with the first. 

275 If the locus of a point is a conic C, the envelope of its polar with 

respect to a conic C is a conic C", and conversely. 

276 The envelope of the base of a triangle inscribed in a conic, and 

two of whose sides pass through fixed points, is a conic. 

277 The locus of the vertex of a triangle which circumscribes a conic 

and whose base angles move along fixed straight lines is a conic. 
Examples. 



CONTENTS XXlll 

CHAPTER XVIII 
Construction of a conic satisfying certain conditions 

ARTICLES 

278 To describe a conic through five points. 

279 To draw the tangent at one of them. 

280 To find the points where the conic meets a given line. 

281 To find the directions of the asymptotes. 

282 To draw the asymptotes. 

283 To describe a conic given five tangents. 

284 To find the point of contact of one of them. 

285 To draw a pair of tangents from a given point. 

286 To describe a conic given four points and a tangent. 

287 To describe a conic given four tangents and a point. 

288 Two properties of a system of conies passing through two given 

points and touching two given lines. 

289 To describe a conic given three points and two tangents. 

290 To describe a conic given three tangents and^two points. 

CHAPTER XIX 

HOMOGRAPHIC generalisation OP CIRCLES AND THE CIRCULAR 
POINTS AT INFINITY, CONICS AND THEIR FOCI, AND OTHER 
ASSOCIATED POINTS AND LINES 

Data 1—23. 
Examples. 

APPENDIX II 
Pascal's Theorem proved for the conic and line-pair by 

THE METHODS OF EuCLID AND ApOLLONIUS 

INDEX 



CHAPTER I 

CROSS-RATIO OF A RANGE OF FOUR POINTS, AND OF A 
PENCIL OF FOUR LINES 

1. The subject treated in the following pages was very fully 
investigated by the ancient Greek geometers except as regards 
its extension to conic sections, where important advances have 
been made in modern times. The old knowledge has been 
revived during the last 100 years, chiefly on the Continent, and 
has been systematised partly by means of taking signs into 
account in dealing with measurements of lengths and angles, and 
partly by means of an improved and powerful notation which 
was rendered possible by this consideration of sign. In the foot- 
notes will be found references (with dates) to the mathematicians 
to whom these improvements are due, and also to the Greek 
geometers. 

This introduction of sign into the consideration of lines and 
angles is one of the distinguishing features of the geometry of 
cross-ratio*. 

Any segment of a line is considered to be positive or negative 
according to the direction in which it is measured, so that, if 
a and h are two points on a straight line, ab = — ha. 

* The first work in which we find the principle of signs systematically 
employed in geometry is Carnot's Geometrie de position (1803). 

M. 1 



2 CROSS-RATIO GEOMETRY [CH. 1 

Again, if (A, B) is the angle between two lines which is 
supposed to be measured by rotation starting from the position A 
towards that of B^ then (^, B) = — {B, A). There is no necessity 
to do more than allude to this, as the student will have met with 
a full treatment of it at an earlier stage of his reading. 

2. Def. a set of points arranged in any manner on a 

J. Axis straight line is called a range, and the straight line 

Base, is called the axis or base of the range. 

A series of concurrent straight lines is called 

Pencil, Centre, ^ pencil, and their common point is called its centre 
Vertex. / 

or vertex. 

The pencils with which we shall deal in the following pages are 
always coplanar. 



Fig. 1. 

Four collinear points taken in pairs give rise to six segments, 
and these segments have sign as well as magnitude. 

An expression such as — ^ : ^-^ is called a cross-ratio of the 

four points or range ahcd, and is the ratio of the distances of the 
Cross-ratio of point a from c and d divided by the ratio of the 
four points. distances of the point h from the same two points, 
and this ratio of ratios, or cross-ratio, is written (abed)*. 

* The term anharmonic function or ratio was given to this expression 
by Chasles in his Apergu historique, 1837, but the term cross-ratio, i.e. ratio 
of the ratios in which cd is divided by a and b, was introduced by Clifford in 
1878, and is now generally adopted. The notation (a, b, c, d) to denote the 
cross-ratio of four points was employed by Mobius, 1827, but was not 
adopted by Chasles in his Geovietrie Superieure, 1852, although in his Traite 
des Sectiom Coniques, 1865, he made use of it and of the corresponding 
notation P [a, b, c, d) to denote the cross-ratio of a pencil of four rays. As 
no advantage is gained by retaining the commas here, we have omitted 




2-3] OF A RANGE OF FOUR POINTS 3 

3. By varying the order in which the four points are taken 
we can form 24 cross-ratios, which, however, are not all different 
in value. For consider 

ac he 

.- , . hd ad ac he 
(hade) = Y- : — = — , S r-y = {(^hcd), 
' he ac ad hd ^ 

, -, ,. ea da ac he , . ,. 

/J z- X ^^ ^^ ac he . 

(dcha) = --: — = — - : — - ^ (abed), 
^ ' da ca ad hd ' 

i.e. the value of a cross-ratio of four points will he unaltered if we 
interchange the j)osit%ons of any pair of points in it, provided 
we also interchange tL. positions of the other pair. 

It should be noticed that in the above equal cross-ratios the 
pair of points a and h are associated together, as are also the pair 
€ and d. It does not matter which pair is mentioned first ; all 
that does matter is that if a and h are interchanged, so must also 
c and d. If we interchange the order of one pair only, we invert 
the cross-ratio, as shewn below. If we pair the points differently, 
we get entirely different cross-ratios, as the student may easily 
see for himself by trial. The relations between them will be 
given in Art. 4. 

Since (hacd) = .— ^ : — ^ = .-^ — ^ , we see that if we interchange 
^ ' hd ad (abed)' ^ 

separately either the first or last pair of points in a cross-ratio, we 

shall obtain another cross-ratio which is the reciprocal of the 

former. 

them, and written the functions simply {abed) and P {abed), reserving 
the commas for use in the notation for involution. If the student uses 
commas at all, it is best to use one comma only, to separate the pairs, 
thus {ab, cd). 

1—2 



4 CROSS-KATIO GEOMETRY [CH. I 

Consequently, the 24 cross-ratios may be arranged in six 
groups of four mutually equal cross-ratios, which may be written 

(abed), (acdb), (adhc), 
and (abdc), (acbd), (adcb), 

of which the first three are formed by the cyclical interchange of 
the letters b, c, d, and the last three are respectively the re- 
ciprocals of the first three. 

Relations between the cross-ratios of four given 

points. 

4. There is an important fundamental relation, discovered 

by Euler in 1747, between the segments of a line made by four 

points, a, b, c, d, on it, viz. 

ab .cd + ac . db + ad . bc = 0, 

in which the second factors of the terms are formed by the 

cyclical interchange of the letters b, c, c?, viz. cd, db, be, and the 

first factors are the distances of a from the remaining points. 

We will prove this and then apply it to find the relations 

between the various cross-ratios of the four points. 

In Fig. 1, if a, b, c, d are the four points 

cd — ad — ac, db = ab — ad, be = ac — ab, 

.'. ab . cd + ac . db + ad . be 

= ab (ad — ac) + ac (ab — ad) + ad (ac - ab) 

= 0. 

Dividing each term by ad . be we have 

ab .cd acbd _ „ 
+ 1=0. 

•• («cc^*)-(«*^'^)-i=o (!)• i 

or, if we denote the three cross-ratios [abed), (acdb), (adbc) by 
X, y, z, (1) becomes 

a; + 1 = 0, . . - = 1 - £t', . . y = -^ . 

y y ^-^ 



ad . cb ad . be 
1 




4-5] OF A RANGE OB^ FOUR POINTS 5 

1 1 x—l 
Similarly - = 1 - 3/, and - = 1 - 0, .*. z = ' . 

Hence, if the value of one of the cross-ratios is given, we can 
at once obtain the values of the others. That is, if x is the value 
of any one of them, the six different values to be found amongst 
the 24 cross-ratios will be 

1 x-\ 

^' r^' X ' 

1 1 ^_ 

x' •'' x-\' 

5. A little consideration of the three expressions 
1 a;~l 

""' r^' IT' 

will shew the student that two of them are always positive and 
the third negative, or this may be seen geometrically as follows : 



-^ $ r- 

Fig. 2. 

Let P, Q, B be three points on a line such that ^-y = x. Then 

l_,-_«^ and J— -^ 
1 X- p^, and j_^- ^^, 

, x-\ . I ^ PQ QR 

Therefore the six cross-ratios are represented by 

PR QP RQ 
PQ' QP' RP' 

PQ QK RP 
PR' QP* RQ' 

where in each line the numerators and denominators of the 



6 CROSS-RATIO GEOMETRY [CH. I 

second and third fractions are formed by cyclical changes from 
those of the first fraction. 

It is obvious from the figure that two of these six ratios are 
negative, viz. the two in which the middle letter of the three 
P, Q, R comes first, whilst the other four are evidently positive. 

6. This geometrical connection between the cross-ratios of 
four coUinear points and the simple ratios of the ^ segments 
formed by three collinear points will probably seem at present 
artificial, but we shall shortly see that it is an immediate con- 
sequence of a very important property of cross-ratios. Meanwhile 
the student may make use of it to deduce the remaining five cross- 
ratios of four points when one of them is known. 

Ex. 1. If ah = 2", &c = 3", cd = l", find (abed), and deduce the values of 
the other five cross-ratios. Ans. V; -9. tV> t*t> -i' l^* 

Ex. 2. Verify Euler's Theorem in the case of the range in Ex. 1. 

7. Seeing then that the values of all the different cross- 
ratios of four collinear points can be expressed in terms of any 
one of the cross-ratios, it will be found convenient, and it will fix 
our attention in any investigation if we select any one of the 
cross-ratios and speak of it as the cross-ratio of the four points 
that we are considering. It is of course immaterial which of the 
24 cross-ratios we take as our standard, and different writers 
have adopted different orders of the letters. All that is necessary 
is the observance of consistency. We have invariably adopted 

the order —j'j-7i and our reason for doing so is that it is the 

arrangement adopted by Chasles. 

8. If no two of the four points coincide^ a cross-ratio cannot 
have as its value 0, +1, or oo ^ though it is capable of assuming 
any other value. 

For, taking the cross-ratio to be — ,: 7-,, if / , = 0, then 
° ad ha ad .he 

either a and c, or h and d coincide. 



i-9] 



OF A RANGE OF FOUR POINTS 



If — ,^^-v- = QO , either a and d^ or b and c coincide. 



ad . he 



If 



ac . hd 



he 



ae 



ah 



ac _ 

' ad hd ad — ah 



ad . be 

.'. ah {ad - ac) = 0, .'. ah.cd = 0, 

'consequently, either a and b, or c and d coincide. 

The relations of Art. 4 shew that the cross-ratios become in 
pairs 0, +1, or oo , if one of them does, as is also shewn by the 
geometrical relations of Art. 5. 

9. Given the cross-ratio (A) of four collinear points a, 6, c, c?, 
of which three are given in position, it is required to find the 
position of the fourth. 




Suppose the three given points are a, b, c. Through a draw 
any straight line aa making any angle with the given line abc, 

and on it take two points a, a such that — = A.. Join a'c. 

aa 

Through b draw b^ parallel to aa, meeting ac in /?. Join a/3 
meeting the line abc in d. Then d shall be the point required. 
For by similar triangles 

ac aa ad aa 

Vc^b^' bd^W 
ac ad aa 



Hence 



he ' hd aa 



\. 



In the above construction X is supposed to be positive. If 
it is negative, we must take a and a on opposite sides of a. 



8 CROSS-RATIO GEOMETRY [CH. I 

10. It still remains to be proved that the point d is 

^ ad \ ac , . 

unique. We nave jr^ = t - t- = x suppose, where k is constant. 

If possible let d' be another point on the line ab such that 
ad' 

= K. 



hd' 



Then 



ad' 

hd' 



ad ad' - dd' 



hd hd'-dd'' 

Therefore either ad' = hd', and consequently ah = 0, or else 
dd' = 0. Now a does not coincide with 6, therefore d' must 
coincide with d. 

11. Remembering that A. is only one of the cross-ratios of 




Fig. 4. 



the four points a, h, c, d, i.e. {ahcd) = \= — , we can shew on Fig. 4 

aoL 

the values of the other cross-ratios {achd) and (adbc). 

Through c and d draw cy, dS parallel to aa. Let ac meet dS 

in 8, and let ad meet cy in y. Let by and bS meet aa in a", a". 



OF A RANGE OF FOUR POINTS 



10-12] 
Then 



so that if aa is taken as the unit of length, a<x, aa", and aa" will 
represent the values of (abed), (achd) and (adbc). 

12. Again, describe a semicircle on aa as diameter, draw 
a'P perpendicular to aa, and join Pa, Pa. 



(-M) = J: 


ad aa' 
cd Cy 


aa aa" 
Cy aa 


(adbc)^f^ 


ac aa!" 
' dc~ d8 


aa aa!" 
' d8~ aa 



Then 



(abed) 



aa 
aa 



aa 



aP 



= cos^ 0. 



Therefore by Art. 4, the six cross-ratios of any four collinear 
points can be represented by 

cos^ Oy cosec^ 0, — tan^ 6, 



sec^^, sin2(9. 



cot^ 0. 



This also follows from the fractions of Art. 5. For if we take 
three collinear points in the order PQP as in Fig. 2, and draw a 
semicircle on PP as diameter, and draw the perpendicular QS 
cutting the semicircle in S, and join SP, then if the angle 
SPP = 6f the values of the six ratios are cos^6' &c. as above. 

If we take the angle (f> at A' we shall obtain the same results 
but in different order, as the two angles and <f) are comple- 
mentary. 

It will be noted that the negative ratios are - tan^^ and — cot- 0, 
of which one is > - 1, and the other < — 1, while of the four 
positive ratios two must be > 1 and two < 1. 

The student should also notice that the points P, Q, R are 
the same as the points a, a, a of Art. 1 1 and Fig. 4. 

The method of Art. 5 is most suitable for expressing the six 
cross-ratios of four points as vulgar fractions, whilst by Art. 1 1 
or 12 (see Art. 13 below), we can most easily express them as 
decimals. 



10 



CROSS-RATIO GEOMETRY 



[CH. I 



13. Suppose now the range abed is given, and we wish to 
find geometrically the value of its cross-ratio. 

In Fig. 3, through a and h draw the parallels aa, 6/3, and on 
the first take aa of unit length. Join ad cutting 6y8 in ^. Join 
CjS, cutting aa in a. Then aa is the cross-ratio required. 

14. Dep. If we consider the four concurrent straight lines 

OA, OB, OC, ODj we define the compound ratio 
Cross-ratio of • / a r\ ■ ( n r\ 
four lines. ^!^ ^^7 ^/ : ^l"" f ' ^^ formed by taking the sines 

sm (A, D) sm (B, D) j & 

of four of the six angles which these lines make with one another 
as the cross-ratio of the pencil (ABCD). See also Art. 16. 

15. If the pencil (A BCD) is cut hy a transversal in the 
four points a, b, c, d, the cross-ratio of the pencil will be equal 

to that of the range abed, and will have the same sign. 




Fig. 5. 



For 



Similarly 



sin (A, C) 



ac 



sm c 

sin (A, C) _ sin c 
sin {A, D) sine? 

sin (B, G) _ sin c 
sin (B, D) ~ sin d 

sin {A, G) sin (B, G) 



sin {A, D) ad 
sin d aO 



ac 
ad' 

be 
bd' 

ac be 



sin {A, D) ' sin {B, D) ad' bd 



13-17] OF A PENCIL OF FOUE LINES 11 



rl3-] 
Now if we suppose the usual convention to hold respecting 
the positive and negative directions of rotation in the description 
of angles, so that sin (A, C) = — sin (C, ^), &c., it is clear that the 
cross-ratio of the pencil (A BCD) is equal to the cross-ratio of the 
range abed, both in magnitude and in sign. 

16. We may remark that the definition given in Art, 14 is 
not often used, and in view of the property proved in Art. 15 it 
would be better to substitute the following : 

Dbf. The cross-ratio of a pencil of four rays is that of the 
range which it forms on any transversal. 

As in Art. 2, when we refer to the cross-ratio of the pencil 
(ABCD), we shall speak of it either as the pencil (ABCD), 
or simply (ABCD). 

The conclusions of Arts. 3 — 8 respecting the cross-ratios of 
four points will apply equally to the cross-ratios of a pencil of 
four rays, and need not be repeated. 

17. Given X, the cross-ratio of four rays OA, OB, OC, OD of 
a pencil of which the first three rays OA, OB, OC are given in 
position, it is required to find the fourth ray. 

In Fig. 6 let any transversal meet the three given rays in 
a, b, c. 




12 CROSS-RATIO GEOMETRY [CH. 

On aO take the point a' so that -^ - \. Join a'c, and through 

h draw 6y8 parallel to aO, cutting ixc in ^. Join 0^. This is 
the fourth ray required. 

Produce 0/3 to cut ahc in o?. 

_, ac be ac ad aa! aO aa 

^''kr'bc 'hd^bp ''b^^^^ 

It should be noted that the above construction is the same as 
that given in Art. 9. 

18. Menelaus' Theorem. If any transversal cuts the sides 
of a triangle ABC in the points a, b, c, 

Ab.Ca.Bc=-bC.aB.cA. 



m 




Fig. 7. 

Through b draw bb' parallel to BC, and join bB. 
Then by Art. 1 5, the range (cb'A B) = the pencil b (cb'AB) 

= the range (a oo CB). 

^, , cA b'A aC ooC 

Therefore -^ : jrh = -^b • — d • 

cB b B aB CO B 

b'A bA , ^C , 

^^^ FB = bC^ ^'''^ ^^^- 

cA bA aC 
" dB'W ^oB' 

Hence Ab .Ca.Bc — — bC. aB . cA. 



18] MENELAUS' THEOREM 13 

Conversely, if three points a, h, c on the sides of a triangle 
ABC satisfy the relation Ah .Ga. Bc — — hC. aB . cA, they are 
collinear. For suppose that he when produced meets BC in a'. 
Then by the above, 

Ah.Ca .Bc = -hC.aB.cA, 
and by hypothesis 

Ah.Ca,Bc = -hC.aB.cA. 
Therefore Ca* : Ca = a'B : aB, 

Ca' : a'B = Ca : aB, 
CB : a'B = CB : aB. 
Consequently the point a' coincides with a. 

Note. Menelaus was a Greek geometer and astronomer, a 
native of Alexandria. He was at Rome studying astronomy in 
the first year of Trajan, a.d. 93. The theorem in the text is 
given in his treatise on Spherical Trigonometry in 3 books, 
which survived in Arabic, and of which a Latin translation 
was first published in a Collection of Greek Geometers made 
at Paris in 1626. 

EXAMPLES. 

1. Any two transversals cut the sides of a triangle in the points P, Q, R 
and F, Q', R'. Prove that 

{BCPP') {CA QQ') {ABRR') = 1. 
Expand and use Menelaus' Theorem. 

2. If a transversal meets the consecutive sides of a polygon ABCD ... 
in the points a, 6, c ... , shew that 

aA.hB.cG ...=aB.hC ,cB .... 

3. Shew that (1) {PQRT)x{PQTS) = {PQRS). 

(2) (PTRS) X {TQRS) = (PQRS). 
[The student should notice the position of the element T in the factors.] 

4. If one of the cross-ratios of a range = -1, find the values of the 
other cross-ratios. 



CHAPTER II 

EQUICROSS RANGES AND EQUICROSS PENCILS. 
PERSPECTIVE 

19. Given a range of Jour points abed on one straight line, and 
a range of four points a'b'c'd' respectively corresponding to them on 
another straight line, if a cross-ratio of the first range is equal to 
the corresponding cross-ratio of the second, then the other cross- 
ratios of the first range are respectively equal to the corresponding 
cross-ratios of the second. 

For suppose (abed) = (a'b'c'd'). 

Then by Art. 4 (1), 

and (a'b'c'd') =: I -.-^^^. 

^ ' (acdb) 

Therefore (acdb) = (a'c'd'b'). 

Similarly (adbc) = (a'd'b'c). 

20. It follows at once from Art. 4 that if two ranges of four 
points have a cross-ratio of the one equal to a cross-ratio of the 
other, then each one of the 24 cross-ratios of the one is equal to 
the corresponding cross-ratio of the other ; and consequently we 
may briefly say that two such ranges have their cross-ratios equal, 
or have the same cross-ratios, or we may speak of them still more 
briefly as equicross, or even as equal ranges. 



19-21] EQUICROSS RANGES 15 

If we have two ranges whose corresponding segments are 
proportional, we shall call them similar ranges, and if the 
corresponding segments are equal, the ranges are said to be 
identical, and are then superposable. All such ranges are, of 
course, equicross. 

Ex. In the range {abed) a& = 3cm., 6c = 2cm., cd=lcm. In an equi- 
cross range {a'b'c'd') find the position of d' when 

(1) a'6' = 2cm., 6'c'=3cm. (2) a'b' = 5 cm., 6'c' = 6cm. 

(3) a'6' = 4cm., 6'c' = 3cm. (4) a'b' = 6 em., b'c' = 4:cm. 

Ans. (1) c'd' = 3; (2) c'd' = 4f ; (3) c'd' = lj%; (4) c'd' = 2. 

It should be noticed that ranges which are equicross are not 
usually similarly divided, though of course they may be so. Thus 
comparing the range (abed) with the equal ranges (1), (2), (3), (4), 
we see that only in the last are the segments proportional to 
those of (abed). 

Ranges in Perspective. 

21. If a pencil is cut by two transversals in the points abed, 
a'b'c'd', the ranges are equicross by Art. 15, for each of them is 
equal to the cross- ratio of the pencil. Hence 

A pencil cuts any two transversals in equicross ranges*. 

This is the fundamental proposition of the subject, and is a 
projective property. It is important to notice that while the 
ratio of the segments into which a transversal is divided by a 
pencil of three rays is the same only for parallel transversals and 
for a pencil of parallel rays, the cross-ratio for a pencil of fotir 
rays is the same for all transversals. 

Def. When two equicross ranges are so placed that they are 
Perspective transversals of the same pencil, i.e. when the lines 

Centre of joining corresponding points on the ranges are 

Perspective. concurrent, the ranges are said to be in perspective, 
and the vertex of the pencil is called the centre of perspective. 

* Pappus, Bk VII, Prop. 129. 



16 CROSS-RATIO GEOMETRY [CH. II 

If the ranges are in perspective, they are necessarily equi- 
cross, but the converse, as we shall see later, is by no means 
necessarily true. What happens when they are not in perspective 
is discussed in Chapter XI. We can, however, always place 
equicross ranges so that they shall be in perspective; a simple 
method of doing this is given in Art. 23. 

If the two ranges when in perspective are parallel it is easy to 
see that they are similar, and conversely, to put similar ranges 
in perspective all that is needed is to make them parallel. 

If two ranges are identical they will also be in perspective 
when they are parallel, the centre of perspective being at 
infinity. 

All this was well known to and discussed by the ancient 
Geometers, as the student will see by referring to Appendix I. 

22. Given two equicross ranges, if the lines joining 3 pairs of 
corresponding points pass through a point, the line joining the 
fourth pair will also pass through the same point. 




Fig. 8. 

Let (abed) and (a'b'c'd') be two equicross ranges, and let aa\ 
bb', cc' meet in 0. If Od does not pass through d', let it meet the 
line ab'c in d". 

Then (a'b'c'd") = (abed) by Art. 21 

= (a'b'c'd') by hypothesis. 
Therefore, by Art. 10, d" coincides with d'. 



I 



22-24] PERSPECTIVE 17 

23. Def. In two equicross ranges, if a point of one coin- 
Common cides with the corresponding point of the other, 
point. the two ranges are said to have a common point. 

If two equicross ranges have a common pointy then the straight 
lines joining the other pairs of corresponding points are concurrent. 



O 




Fig. 9. 

Let the points a, a' coincide, and let hh\ cc' meet in 0. 

Join Oa, and let Od meet a'h'c' in d". 

Then {abcd)^(a'b'c'd") by Art. 21 

= {a'h'c' d') by hypothesis. 

Therefore, by Art. 10, d" coincides with d'. 

Note. When this is the case the ranges are in perspective, 
centre 0, but of course they may be in perspective even if (a, a') 
do not coincide, but it is not so easy to put them in perspective in 
that case. This will be discussed when we come to homographic 
ranges in which there are more than 4 points on each range. 

Pencils in Perspective. 

24. IJ two pencils are subtended by the same ramge, they are 
equicross. 

By Art. 15 the cross-ratios of each of the pencils are equal to 
the cross-ratios of the range abed. 

Def. When two equicross pencils are so placed that they 
Perspective subtend the same range, i.e. so that corresponding 

Axis of rays intersect in collinear points, the pencils are 

Perspective. gj^j^ ^^ y^g ^^ perspective, and the common range 

M. 2 



18 



CROSS-RATIO GEOMETRY 



[CH. II 



is called the axis of perspective. A simple method of putting 
them in perspective is given in Art. 25. 




Just as in the case of ranges, if two pencils are equicross, 
they are not necessarily in perspective, and what happens in 
general is discussed in a later chapter. 

25. Def. In two equicross pencils if a ray of one pencil 

coincides with the corresponding ray in the other, 

the two pencils are said to have a common tdy. 

If two equicross pencils of 4 corresponding rays have a 

common ray^ then the other pairs of corresponding rays will 

intersect in three points which are collinear^ and conversely. 




Let 
Oc, O'd, 
comTHon 



Fig. 11. 

the rays of the pencils be Oa, Oh, Oc, Od, and O'a, O'h, 
so that the rays 0«, O'a coincide. Join he meeting the 
ray in a, and let he meet the rays Od, O'd in 8, 8'. Then 



A 



25-26] TRIANGLES IN PERSPECTIVE 19 

since the two pencils are equicross, (abcS) = {abch'), and therefore 
by Art. 10 the points 8, h' coincide, i.e. the line he passes through d. 

Conversely, if two pencils are such that the intersections oj 
three pairs of corresponding rays are collinear, and the fourth 
pair of rays are in the same straight line, the pencils are equicross. 

For in Fig. 11 suppose the point d, to lie on the line he. 
Then by Art. 24 the pencils 0(ahcd) and 0' (ahcd) are equicross. 

If two equicross pencils with centres 0, 0\ are such that the 
intersections of three pairs of corresponding rays are collinear, as 
a, 6, c in Fig. 10, then the fourth pair of rays tvill also intersect 
on the line abc, for if they meet this line in two separate points 
8, 8', we should have (abcS) = (abcS'), and therefore, by Art. 10, 
8 and 8' must coincide. 

In all the above cases the pencils are in perspective, and the 
line on which the pairs of corresponding rays intersect is the axis 
of perspective. 

Triangles in Perspective. 

26. Def. If two triangles are such that the lines joining 
Co-polar *'^® pairs of corresponding vertices are concurrent. 

Triangles. the triangles are said to be co-polar. 

If two triangles are such that the intersections 
Co-axial of pairs of corresponding sides are coUinear, the tri- 

Triangles. angles are said to be co-axial. 

Two tria7igles ivhich are co-polar are also co-axial, and two 
triangles which are co-axial are also co-polar. 

Let abc, a'b'c be two triangles such that the lines aa, hb', cc 
meet in the point 0. Let {be, h'c) meet in a, (ca, c'a) in y8 and 
{ah, ah') in y. It is required to prove that a, fi, y are coUinear. 

Let the line Occ meet ab in 8 and ah' in 8'. 

Then by Art. 21 the range (a86y) = the range (a'8'6'y), and by 
Art. 15 the pencil c {ahby) = the pencil c'{a'8'b'y). 

Now the corresponding rays c8 and c'S' are also common rays. 



20 



CROSS-RATIO GEOMETRY 



[CH. II 



and therefore by Art. 25 the intersections of {ca, ca\ (cb, c'b'), 
(cy, c'y) are collinear, i.e. the points /?, a, y are collinear. 

Conversely, let the points a, /3, y be collinear. It is required 
to shew that aa', bb', cc' are concurrent. 

Since the pencils c (aBby) and c'(a'B'b'y) are such that the 
intersections of three pairs of corresponding rays are collinear, 
and the fourth pair of rays are common, the pencils are equicross^ 
by Art. 25. 

y 




I 



Fig. 12. 



Therefore c (aSby) = c{a'S'b'y), 

.-. by Art. 15 (ahby) = (a'S'b'y), 

Perspective, ^^^^ ^Y ^^^' ^^j '^^'j ^^ ' ^^' ^^^ concurrent. 

Two triangles which have the above properties 
are said to be in per.^pective. The point is called 
the pole, or centre of perspective, and the line a^y 
the axis of perspective. 



Pole, 
Centre of 
Perspective, 
Axis of 
Perspective. 



Ex. 1. In Fig. 8. If the ranges intersect in e shew that the range 
formed from any four of the points a, b, c, d, e is equicross with the 
corresponding range on the other transversal. 

Ex. 2. If two pencils O (abed), 0' {abed) are in perspective shew that 
the pencil formed from any four of the rays 0{0'abcd) is equicross with 
the corresponding pencil from O'. 



CHAPTER III 

HARMONIC RATIO 

27. Def. If the points a, a divide a segment ef internally 
. and externally in a given ratio, i.e. in such a way 

Range, ^j^^^ ^j^^ cross-ratio (aaef), i.e. — • ,-.= -1, the four 

Harmonic \ ^ /' of aj 

onjugates. ^^^ g^-^ ^^ form a harmonic range 'j the points a, a 
are said to divide the segment ef harmonically, and are called the 

harmonic conjugates of the points e^f. 

/• 

Since the above relation may be written — r : -r-, == — 1, we see 

^ ea fa 

also that the points e, / divide the segment aa harmonically, and 
that they are the harmonic conjugates of the points a, a. 

The points a, a may be spoken of as a pair of conjugate 

points, and similarly for e, f Also each point of a 

Harmonic, V^^^ ^^ conjugates is called the fourth harmonic, or 

Harmonic the harmoriic conjugate of the other for the second 

conjugate. ^^.^ ^^ ^^^^^^ 

28. Substituting the value — 1 in any one of the six 
expressions in Art. 4 we see that when a range is harmonic the 
six values of the cross-ratios reduce to three, viz. - 1, J, 2. 

Conversely, if a range has one of its cross-ratios equal to 
either — 1, or i, or 2, the range is harmonic. 

Since - 1 is the value of the cross-ratio (aa'e/), where aa and 
ef are pairs of conjugate points, the reciprocal of this ratio, i.e. 
(aafe), see Art. 3, must also =—1. 



22 CROSS-RATIO GEOMETRY [CH. Ill 

Hence {aa'ef) = (aafe) = (a'aef) by Art. 3. Consequently, a 
range is harmonic if it has a cross-ratio whose value is unaltered 
on interchanging the positions of a single pair of points. When 
this is the case, the pair of points so interchanged are conjugates. 

We may remark that not only is (aa'ef) = {alaef), but it 

follows that 

• {aeaf) = {a'eaf) - {afa'e), 

and {aefa) = {a'efa) = {afea), 

i.e. in a harmonic range each of either pair of conjugate points is 
interchangeable in any of the cross-ratios, not only in the funda- 
mental cross-ratio whose value is — 1. 

If we take the other values of the cross-ratios, viz. {aeaf) = 2, 
we have 

aa' ea . / /• « ^ ' 

-— ;:-2= = 2, .. aa .ef=2aj .ea, 

«/ «/ 

and (a/ea') = J , — 7 : ^, = J , .. aa! . ef- 2ae . af 

aa Ja " 

These results can also be easily obtained from Euler's Equa- 
tion, Art. 4, 

ad . ef+ ae.fa'+ af . a'e = 0. 

^ . / , /.N 1 . ae, af , 

For since (aa e/ ) — — 1, i.e. —^ — r = — !> ^ 
^ -^ ' af. a e 

therefore aa! . ef= '2ae . af= — 2af . a'e. 

Ex. If aa' = 4 cm. and is divided by e and / internally and externally in 
the ratio 3 : 1, find the values of aa' .ef, ae.a'f and af .a'e, and verify the^ 
above relations. 

29. To find the fourth harmonic of three given points. 

We will employ the method of Art. 9. See also Art. 32, Cor.j 

(1) Let the segment aa' be cut internally at the point e. 1\ 
is required to find the fourth harmonic of e for the points a, a' 

Through a and a' draw any two parallel straight lines aa, «'/?,] 
and on aa. take two points a, a' such that aa = — ad. 



,29-30] HARMONIC RATIO 23 

I 

' Draw ae meeting a^ in /?. Join ap meeting aa' in f. Then 
/ will be the point required. 

For ae : ea = aa : fta 

= a a : pa 

^af'.a'f. 




Fig. 13. 

(2) Suppose the segment aa is cut externally at f. As 
before, take aa = — aa. Join a!f cutting the parallel a'p in jB. 
Then a^ will meet aa in the required point e. 

CoR. If one of the pair of conjugate points ef is at infinity, 
as f suppose, then e is the mid-point of aa, as is obvious from 
the construction in Fig. 1 3, since af will then be parallel to aa\ 
and aft will = W — — aa. 

This also follows from the algebraical relation {aa'ef) = — 1, 

/» 

as the student should verify, noting that ^=1, and therefore 

ay 

-r = — 1 • Hence 
ae 

If one of the four points of a harmonic range is at infinity, its 

conjugate is at the mid-point between the other two, and vice versa. 

30. Given three rays of a pencil, to find the fourth harmonic 
of one of them for the other two. 

Let PA, PB be two of the given rays, PC the third. It is 
required to find the fourth harmonic of PC for PA and PB. 



1 



24 CROSS-RATIO GEOMETRY [CH. 

Draw any line ah parallel to PC meeting PA, PB or these 
lines produced in a, 6, and bisect ah in d. Then Pd is the ray 
required. 

For since ah is bisected at d, the range {ah qo d) is harmonic, 




Fig. 14. 

by Art. 29, and consequently by Art. 15 the pencil P {ABCD) is 
harmonic. 

Cor. From the above construction it is evident that if one 
of the rays PC, PD bisects the interior angle between PA and 
PB, its conjugate will bisect the exterior angle between them. 
Hence, 

If one of three rays of a pencil hisects the angle hetween the 
other two, its conjugate is at right angles to it. 

31. Every harmoriic range determines a harmonic pencil at 
every centre, and every harmonic pencil determines a harmonic 
range on every transversal. See Arts. 15 and 21. 



31-83] 



HARMONIC RATIO 



25 



Relations between the Segments of a Harmonic Range. 

32. Let aa\ ef be two pairs of conjugate points forming a 
harmonic range, and let 0, 0' be the mid-points of aa' and ef 
respectively. 

O O' 



a' 



f 



Fig. 15. 



Then 06.0f^0a?=0a\ 

■n. . ae ae ^^ 

r or since — ^ + -7>== U *, 

af af 

Oe-Oa_ Oe-O a _ Oe + Oa 
" Of^~Oa~~' Of^Ja'~~ Of+Oa' 
whence, clearing of fractions, we have at once 

20e.Of=20a\ 
the other terms cancelling. 

.'. Oe. Of= Oa' = Oa' = (| aa'f *. 
Similarly O'a . O'a' = (^ eff^. 

Cor. This gives rise to another construction for the fourth 
harmonic. 

On aa as diameter describe a circle, and let aa' be the 
diameter perpendicular to aa. Let ae meet the circle in P. 
Then a!P will meet aa in the required point f. This is obvious 
from consideration of similar triangles. 

33. Another interesting property is that 

1 i -_i 
ae af aa ' 

-r, . o^ a'e - 

± or since — + — =0, 

«/ a/ 

* Pappus, ]5k VII, Lemma xxxiv. 

t Pappus, Bk VII, Lemmas xxvi, xxvii. 



26 



CROSS-RATIO GEOMETRY 



[CH. Ill 



ae ae — aa 
af af— aa 



0, 



.'. aa' . a/+ aa . ae = 2ae . a/, 

• i_ 1 - A 
' ' ae af aob ' 

This property shews that aa! is the harmonic mean between 
the segments ae, af^ and may be considered the reason for the 
name ' harmonic ' being applied to the range. 

34. The following 7 relations are given by Pappus, Bk vii, 
in his Lemmas to Euclid's Porisms, and are left as exercises to 
the student. 

Lemmas xxii and xxiv, 
xxiii and xxv. 



(1) 

(2) 
(3) 



(4) 

(5) 
(6) 

(7) 
(8) 



xxvi and xxvii. 



Lemma xxxi^ 



00'2-aO-^ + eO'^ 

ae'^laO'.Oe 

ea'=2aO'.Oe 

ae^ aO' 

ea"^ a'O' 

ae . ea = Oe . ef^ 

af:a'f=0/.e/ 

ae of 

ea' af 

Shew that if the cross- ratio {aa'ef) is equal to either 
- \ (aeaf) or — 2 {afea) the range aa'ef is harmonic, aa' and ej 
being pairs of conjugate points. 

35. Given two segments aa', hh' on a line, it is required to 
find on the same line a point such that Oa . Oa' ~ Ob . Ob'. 
Assuming the existence of such a point, we have 

Oa . Oa = Ob . Ob\ '^ 

Oa _0b^ _ Ob' -Oa ab^ 
" Ob~0^'~ OaT^Ob ~ ba' ' 

Hence the construction. Through a and b draw two parallel 
lines, and on them take aa = ab\ and b/3 = ba'. Then the point 
where a/3 meets the given line is the required point 0. 



A 



r 

■34-3 



34-36] HARMONIC RATIO 27 

If the segments aa\ bb' do not overlap, the products Oa . Oa 
and Ob . Ob' are both positive, and we may put them = Oe^ or Of^. 
Then each of the segments aa', bb' will be divided harmonically by 
the real points e, f. 

If the segments aa', bb' overlap, the products Oa . Oa and 
Ob . Ob' are both negative, and consequently the points e, f are 
imaginary, but we shall still say that the segments aa', bb' are 
divided harmonically by the imaginary points e, /. 





a b O a' b' 

Fig. 17. 

It is evident from the construction that for any given position 
of the segments aa, bb' there is one and only one position of the 
point 0, and only one position of the segment ef. Hence 

Only one segment can be found to divide two given collinear 
segments harmonically. 

36. Given a pencil of four rays, if we take the rays in two 
pairs in any manner, then in each arrangement we can always 
find a third pair of rays which will form a harmonic pencil with 
each of the given pairs. 

This follows at once from the preceding Art. by drawing 
a transversal cutting the arranged pairs in aa', bb', and finding 
the harmonic conjugates e,foi their segments. Then if is the 
vertex of the given pencil, Oe, O/ave conjugates for Oa, Oa'^ and 
also for Ob, Ob'. The rays Oe, Of will be real or imaginary 
according as we take the two given pairs to be non-overlapping or 
otherwise. 



CHAPTER IV 

HOMOGRAPHIC RANGES AND HOMOGRAPH IC PENCILS 

37. When two ranges are in perspective we can have as 
many points as we please on each by supposing a ray to rotate 
round the centre of perspective, and so determine a moving point 
on each range such that the two moving points are always a pair 
of corresponding points. The property of such ranges in per- 
spective is that the cross-ratio of any four points on one is equal 
to the cross-ratio of the corresponding four points on the other. 
Now suppose that in this way a whole series of points are fixed 
on the two ranges, and then the ranges are moved away so as 
no longer to be in perspective, without disturbing the relative 
positions of the points on each range*. The quality of cross- 
ratios of course still remains. Such ranges are called homo- 
graphic. In the next article the definition is given more formally, 
and without reference to perspective properties. 

38. Def. If two straight lines are divided at corresponding 
nomographic points in such a manner that the cross-ratio of any 
Ranges. fo^r points of the one is equal to the cross-ratio of 
the four corresponding points of the other, the two straight lin^s 
are said to be divided homographically, and their points of division 
are said to form two homographic ranges. 

* We may even move the ranges so that these lines coincide, having of 
course the ranges distinctly marked to prevent confusion. This case is very 
important, and is discussed later. 



37-40] HOMOGRAPHIC RANGES 29 

It is important to emphasize the fact that when we speak of 
two lines Z, IJ being divided homographically, we mean that 
every point on L belongs to the first range and every point on L 
belongs to the second range, and that each point on either of the 
lines corresponds to one and only one point on the other. This 
correspondence may be arranged in an infinite number of ways by 
taking any three points a, 6, c on Z, and any three a', h\ c' on L' 
as their correspondents. Then, corresponding to any position of a 
variable point m on L, we can find one, and only one point m on 
L' such that {ahem) ~ {a'b'c'm'), for the value of the cross-ratio is 
then fixed, and by Art. 10 the point m' is unique. Of course, 
for every different position of m the cross-ratio (ahem) will have a 
different value, but the points a, h, c and a', h', c' will remain un- 
changed, and will, as it were, determine the character of the 
Character- different cross-ratios. For this reason we shall 

istic of a often refer to the sets of points ahc and a'h'c as the 

^''*^®* characteristics of the ranges L and L'. 

For shortness we shall often denote a range (abede ...) by (a), 
and a pencil P (ahcde ...) by P (a). 

39. Ranges which are homographic to the same range are 
homographic to one another. 

From Def . Art. 38, it follows at once that if a range of points 
(a) is homographic to a range (a'), and also to a range (a"), then 
the cross-ratio of any four points of the range (a) will be equal to 
the cross-ratio of the four corresponding points of the range (a")^ 
and therefore by Art. 38 the ranges (a') and (a") are homo- 
graphic. 

40. Given two lines L, L', one of tvhich, Z, is divided in any 
manner at the points a, h, c, d ..., it is required to find on U 
corresponding points a, h', c', d' ... so that the two lines may he 
divided homographically. 

As pointed out in Art. 38 this may be done in an infinite 
number of ways, because we may select any three points a\ h\ c 



30 



CROSS-RATIO GEOMETRY 



[CH. IV S 



that we please on L\ and take them as the points corresponding 
to rt, 6, c on Z, we can then by Art. 9 find the points c?', e ... 

corresponding to g?, e 

It should be noticed that there is one range, and one only, 
corresponding to each selection of dh'c. 




a' b' c 

Fig. 18. 



One practical way of finding the series of points on L' after 
having determined on the three characteristic points a'h'c is as 
follows : 

Through a draw a line L" making any convenient angle with 
Z, and take ah"= a'h\ ac" -^dc\ and let hh'\ cc" meet in S. Then 
the lines drawn from S to the points d, e ... will meet L" in 
points d'\ e"..., and if on L' we take a'd'=ad", de-ae"...., the 
points d\ e ... will be the points required on L' corresponding to 
the points d', e ... on L. 

That is, the line L" and the range on it are only the line IJ 
and its range moved to a position in which it is in perspective 
with the range on L. See Art. 37. 

In particular we can find the points on each which correspond 
to points at infinity on the other by drawing through aS' lines 
parallel to Z" and Z, meeting Z, Z" in / and J" . The point 



I 

It 7 a 



40-42] HOMOGRAPHIC RANGES 31 

will correspond to the point at infinity on L'\ and therefore also 
to the point at infinity on L', The point J" will correspond to 
the point at infinity on Z, and if on L' we take a! J' = aJ'\ the 
point J' will correspond to the point at infinity on L. See also 
Art. 46, Fig. 20, and Art. 54. 

This construction can be applied to the case where the points 
a', 6', c' are on the line Z, i.e. when the line L' coincides with the 
line X, which is sometimes desirable, and which has been referred 
to in the footnote to Art. 37. 

41. The following results are important and obvious exten- 
sions of the theorems in Arts. 21, 23. 

(1) If two straight lines aL, ah" are cut hy a pencil, they are 
Common divided homographically . The point a evidently 
Point of Two represents two coincident corresponding points, 
Ranges. ^^^^ ^^ g^^-^j ^^ y^^ ^ common point of the two 
ranges on L, L". 

(2) If two straight lines are divided hom,ographically, and if 
their point of intersection is a common point of the ranges, the 
straight lines joining the other pairs of corresponding points are all 
concurrent, and tJie ranges are said to be in perspective. 

42. Def. When two pencils, each containing any number 
nomographic of rays, are such that they have each ray of one 
Pencils. pencil corresponding to a ray of the other in such 
a way that the cross-ratio of any four rays of the one is equal to 
the cross-ratio of the four corresponding rays of the other, the 
pencils are said to be homographic. 

A similar remark to that made in Art. 38 respecting 
homographic ranges might be made here respecting homo- 
graphic pencils. 

By Art. 15 if a pencil is cut by a transversal, it will be 
permissible and convenient to say that the pencil and transversal 
are homographic. 



32 CROSS-RATIO GEOMETRY [CH. IV 

• Def. If two pencils having either the same or different 
Superposable vertices are such that the angles between each pair 
or Identical of rays of the one are equal to the angles taken in 
Pencils. ^^le same sense between the corresponding pairs of 

rays of the other, the pencils are said to be superposable or 
identical. 

Superposable pencils are obviously homographic. 

43. If two pencils 0(ABC ...), 0' {A' B' C . . .) are homographic^ 
and frowj rays are drawn perpendicular to the rays O'A', 0' B' ... 
and forming the pencil (abc ...), and if the pencil 0' {a'h'c' ...) is 

formed from {ABC . ..) ina similar manner, the pencils {abc ...) 
and 0' {a'b'c ...) are homographic. 

For the angle O'Oa is the complement of 00' A', and O'Ob is 
the complement of 00' B\ therefore aOb = A'O'B', &c. Therefore 
the pencils {abc ...), 0' {A' B' C . . .) are superposable. Similarly 
the pencils 0' {a'b'c ...), {ABC ...) are superposable. But 
0{ABC ...) and 0' {A'B'C ...) are homographic, therefore so also 
are {abc ...) and 0' {db'c ...). 

44. Pencils which are homographic to the same pencil are 
homographic to one another. 

From Def. Art. 42 it follows that if a pencil P {a) is homo- 
graphic to a pencil P'{a'), and also to a pencil P"{a"), then the 
cross-ratio of any four rays of the pencil P'{a') will be equal to 
the cross-ratio of the four corresponding rays of the pencil 
P'{a"), and therefore the pencils P'{a') and P'{a") are homo- 
graphic. 

45. If two pencils are drawn from two centres 0, 0', and are 
such that their rays intersect by pairs iri a series of collinear 
points, the pencils are homographic. 

For each of the pencils is homographic with the range on 
their common transversal ahc, Fig. 19. 



48-46] HOMOGRAPHIC PENCILS 38 

If the line 00' be produced to meet the transversal abc in Ky 
Common Ray *^® ^^^^ ^^ ^^^^ evidently represent two coin- 
ofTwo cident corresponding rays, and may be called a 

Pencils. common ray of the two pencils. 




Pencils in 
Perspective 



Conversely, if two homographic pencils have a 
common ray, their other pairs of corresponding rays 
tvill intersect in a series of points which are collinear, jind the 
pencils are said to he in perspective. Art. 25. 

46. To find a range which will he in perspective with each of 
two given homographic ranges which are not in perspective tuith 
each other. 




Fig. 20. 



M. 



34 



CROSS-RATIO GEOMETRY 



[CH. 



Let (abed...) and (a'b'c'd' ...) be the given homograph 
ranges. Then if on the line joining any pair of corresponding 
points a, a' we take two points P, P', and form the pencils 
P {abed ...), P' {a'b'c'd' ...), the pairs of corresponding rays of 
the two pencils will intersect in a series of collinear points 
P, y, 8 ... by Art. 25, for they are homographic, and have a 
common ray. 

47. If we have two homographic pencils P {ABC ...), 
P'{A'B'C' ...), and if through the point of intersection of a pair 
of corresponding rays we draw two transversals meeting the rays 
of the pencils in the two ranges abc ..., a'b'c' . . . , then since the 
ranges are homographic and have a common point, the lines aa', 
bb', cc ... are concurrent by Art. 41 (2). 

48. Given a pencil of four rays, and of « second pencil three 
rays which correspond to three rays of the first pencil, it is required 



1 

3hic ^H 

I 




Fig. 21. 



47-49] HOMOGRAPHIC PENCILS 35 

to find ajourth ray of the second pencil corresponding to the fourth 
ray of the first so that the two pencils may he equicross. 

Let OA, OB, OC, OD be the rays of the first pencil, 0' A\ 0'B\ 
O'C the three rays of the second corresponding to OA, OB, OG. 
It is required to find a fourth ray O'D' so that the pencils 
0{ABGD) and 0'{A'B'C'D') may be equicross. 

Let a be the point where OA, O'A' intersect. Through a 
draw a transversal passing through the point h where OB, O'B' 
intersect, and let it meet 00 in c and OD in d. Through a draw 
a second transversal passing through the point c where OG, O'G' 
intersect, and let it meet O'B' in h' . Then h and h' both lie on 
O'B', and c, c' both lie on OG. Let S be the point where these 
two lines intersect. Join Sd, and let it cut the transversal ab'c 
in d'. Then O'd' is the ray required. 

For (abed) and {ah'c'd') are the ranges in which two trans- 
versals are cut by a pencil, centre aS', and are therefore equicross, 
by Art. 21. 

49. There are two classes of questions in the solution of 
which the properties of homographic ranges or pencils in per- 
spective are immediately applicable, viz. : 

(1) Those in which it is required to prove that the locus of a 
moving point is a straight line ; 

(2) Those in which it is required to prove that a moving 
straight line passes through a fixed point. 

In (1) we obtain two homographic pencils having a common 
ray, viz. the line joining the vertices, and having the different 
positions of the moving point for the intersections of pairs of 
corresponding rays. 

In (2) we obtain two homographic ranges having a common 
point, and having the moving line in its different positions joining 
pairs of corresponding points. 



3—2 



36 CROSS-RATIO GEOMETRY 



EXAMPLES. 




1. A line is drawn in a given direction, and is terminated by two given 
lines. Find the locus of the point which divides it in a given ratio. 

Let the moving line in any position meet the given lines OA, OB in the 

points m, m', and let ni7n' be divided at P so that mP=K .m'P where K is 

constant. Then the different positions of the moving line constitute a 

pencil of parallel rays whose centre is at infinity, and therefore the ranges 

(m), {m') are homographic. Let vi^P^nii be any other position of the moving 

mP 
line. Then the range {mm' Pec ) = —^=K=the range {mimi'Picc ). 

The ranges being equicross, and having go for a common point, are in 
perspective, and therefore PP^, mmi, and m'mi are concurrent, i.e. the locus 
of P is a straight line passing through 0. 

2. Given two lines Oa, Ob, and two fixed points a, b on them, and also 
two variable points m, m' on them such that 0m-\-0m'=0a+0h. Find the 
locus of the intersection of am' and bm. 

By the given condition am = bm', and the ranges (m), {m') being identical 
are homographic. Therefore the pencils b (m), a {m') are homographic. 
And when m is at a, m' is at b, and consequently ab is a common ray and the 
pencils are in perspective. Therefore the required locus is a straight line. 

3. A good illustration of the method is afforded by the following 
proposition, which is the only one of Euclid's Porisms which has come down 
to us in a complete form. See Appendix I. As given by Pappus the 




EXAMPLES 



37 



enunciation is somewhat different, but for our purpose it may be stated as 
follows. 

Given a variable triangle ABC wJiose sides pass through three fixed 
collinear points P, Q, R, If the vertices B and C move along the given lines 
OD, OE, the vertex A will also describe a straight line passing through O. 

By Art. 41 (1) the ranges {B) and (C) are homographic. Therefore by 
Art. 15 and Def. Art. 42 the pencils Q (J5) and B, (C) are homographic, and 
the corresponding rays QD and RE are common rays. 

Hence by Art. 45 the pencils Q (B) and R (C) are in perspective. There- 
fore the point A lies on a fixed straight line. 

Also, since is a common point of the ranges (B) and (C), the locus of A 
passes through 0. 

It is easily seen that the above is equivalent to the property proved in 
Art. 26, viz. Co-axial triangles are co-polar. 

4. In Ex. 3 if the points Q, R instead of being collinear with P, are 
collinear with O, find the locus of A. 

Drawing a figure we see that the ranges (B), (C) are homographic, with 
for a common point, and therefore the pencils Q{B) and R{C) are homo- 
graphic with QOR for common ray, and the pencils are in perspective. 
Hence the point A lies on a straight line, which, however, does not pass 
through 0. The student should draw the line which is the locus of A in 
this case. 

5. Given a variable triangle ABC, two of whose sides pass through the 
fixed points P, Q. If the vertices move along three concurrent lines OD, 
OE, OF, the third side will pass through a fixed point R collinear with P 
and Q. 




Fig. 23. 



38 CROSS-RATIO GEOMETRY [CH. IV 

The ranges (A) and (B), by Art. 41 (1), are each homographic with the 
range (C), and therefore with one another, by Art. 39. 

Also, when G is at 0, A is at 0, and B is at 0, and the ranges (A) and 
(B), having for common point, are in perspective, i.e. the line AB passes 
through a fixed point. 

If PQ meets OA in a, OB in b, and OC in c, then when A is at a, C is at c, 
and B is at &. Therefore PQ is one of the positions of the base, and con- 
sequently the fixed point through which the base passes is collinear with 
P and Q. 

This proposition may be stated Co-polar triangles are co-axial. 

6. In Ex. 5 if the vertex G moves along a straight line which does not 
pass through 0, and if the points P, Q are collinear with 0, shew that AB 
passes through a fixed point. 

Let the line OPQ meet the locus of G in S. Then when A is at 0, G is 
at S, and B is at 0, and the ranges (A) and (B) are in perspective as before. 

If the locus of G meets OA in a and OB in ^, the fixed point R through 
which AB passes is the intersection of Pa, Q^. The student should draw a 
figure and write out the complete proof. 

For an analytical treatment of these examples, see Salmon's Gonics, 
pp. 39—48. 

7. Three points F, G, H are taken on the side BG of a triangle ABG ; 
through G any line is drawn cutting AB and AG in L and M respectively. 
FL and HM intersect in K. Prove that K lies on a fixed straight line 
passing through A. 

Outline of Proof. (L) and (M) are homographic ranges. F(L) and 
H(M) are homographic pencils having FH for common ray. 

8. ABG is an isosceles triangle, and on the equal sides AB, AG equi- 
lateral triangles ABD, AGE are described. BD, GE meet in F, and BE, GD 
meet in Gr. Shew that A, F, G are in a straight line. 

9. A point P, capable of moving along a given straight line, is joined to 
two fixed points B, G, and the lines PB, PG intersect another given straight 
line in X and Y. Prove that the locus of the intersection of BY and GX is 
a straight line. 

10. A, D, G are three fixed points on a given straight line. GE is any 
other fixed line through C, -B is a fixed point, and B is any moving point on 
GE. The lines AE and BD intersect in Q, the lines GQ and DE in R, and 
the lines BR and ^C in P. Prove that P is a fixed point as B moves 
along GE. 



CHAPTER V 

CROSS-AXIS AND CROSS-CENTRE 

50. Given three points a, b, c on a line L, and also three points 
a, b\ c' on a line L\ the points a, ft, y in which the pairs of lines 
(be, b'c), {ca', c'a), (ab', a'b) intersect are collinear. 




Let the lines L, L' intersect in a point which, considered as a 
point on Z, we will denote by p, and considered as a point on L' 
we will denote by q'. Consider the three points a, b', c on L' as 
corresponding to «, 6, c on L. By Art. 40 or 46 find p the point 
on L' corresponding to the point p on L, and find q the point 
on L corresponding to the point q' on U. 

Then {pqab) = (^p'q' a'b'). Therefore the pencils a' (p^a6) and 



40 CROSS-RATIO GEOMETRY [CH. V 

a {p'q'a'h') are equicross, and have the common ray aa, and there- 
fore by Art. 25 the points p', g, y are collinear, i.e. the point y lies 
on the line p'q. By a similar reasoning we see that the points 
a, ft lie on the same line p'q. 

Def. The fixed line p'q is called the homographic 
or cross- axis* of the two ranges. 

51. The proposition of the preceding article is one of 
considerable historical interest. If we join the pairs of points 
in the order a6', 6'c, ca', a'b, he', c'a, we obtain the hexagon 
ab'ca'hc', whose vertices lie by threes on the pair of lines L, L', and 
whose opposite sides, taken in pairs, are {ah', ah), (he, h'c), 
(ca', c'a), of which a, ft, y are the points of intersection. Stated 
in other words, the proposition tells us that if the vertices of a 
hexagon lie by threes on two straight lines, the points in which 
its opposite sides intersect lie on a straight line, being, in effect, 
the Pascal line of the hexagon inscribed in a line pair. This 
property was probably known to Euclid (300 B.C.), and employed 
by him without proof in his Treatise on Porisms. 600 years 
afterwards Pappus supplied a proof depending on Menelaus' 
Theorem. Thirteen centuries afterwards, in 1640, Pascal 
enunciated a similar theorem without proof as a property of 
a hexagon inscribed in a circle, and it was only after another 
interval of 166 years that its correlative was discovered for the 
conic by Brianchon in 1806. 

52. IJ m, m! are any pair of i^oints on the lines L, L' such that 
{ahem) — {a'b'c'm), and i/mc', m'c meet in fx, the point jx will lie on 
the cross-axis. 

For the pencils a [ahem) and a {ah' cm!) have the same cross- 
ratio and the common ray aa. Therefore by Art. 25 their 
corresponding rays intersect on a straight line. Now the pairs of 
rays {ah', ah) and {ac, a'c) intersect on the line yft. Hence the 



Suggested by Dr Filon, Projective Geometry, Pref. v, 1908. 



rf 



I ►1-54] 
ntersec 



CKOSS-AXIS 



41 



ntersection of (cm', cm) lies on the same straight line, which, as 
we have seen, depends solely on the positions of the character- 
istics ahc, a'h'c'. 

By taking (a, a') or (6, h') as centres of the pencils we see 
that the pairs of lines {am\ a'm) and {hm, h'm) also intersect on 
the line a/8y. 

53. The existence of the cross-axis is of supreme importance, 
for it gives us a simple means of dividing a line L' homographic- 
ally to a given divided line L. For we have merely to take any 
three points a, h\ c on L' to correspond to a, b, c any three given 
points on L, and construct the cross-axis. Then to find the 
point on L' corresponding to any point m on L join ma, cutting 
the cross-axis in fi. Then a/x will cut L' in the required point m'. 

54. Bt/ meanti of the cross-axis to jind the 'point J' on U 
corresponding to the point at infinity on L. 




Fig. 25. 

If c, c' are any pair of corresponding points, through c draw a 
line parallel to L meeting the cross-axis p'q in j. Then cj will 
meet L' in J', the point required. 

To find the point I on L corresponding to the point at infinity 
on X', draw ci parallel to L\ meeting the cross-axis in i. Then 
c'i will meet L in the required point /. 



42 



CROSS-RATIO GEOMETRY 



55. The line joining the points /, J', which correspond to the 
points at infinity^ is parallel to the cross-axis. 

Denoting the points at infinity on L and L by oo and oo ', 
since 

{pqicc ) = {pq'zo V), 

•• ql p'J'^ 
.'. IJ' is parallel to p'q. 

This is also obvious from Art. 52 which shews that IJ' 
and oo'oo intersect on the cross-axis. 

56. The points / and J' can also be found by .moving one of 
the lines, as Z', parallel to and along itself until the point p' 
coincides with its correspondent /;. The ranges are then in 




Fig. 26. 



perspective, Art. 41 (2), and if S is the centre of perspective, 
/ and J' are obtained by drawing through S parallels to L' ^^ 
and L. Cf. Art. 40 and Fig. 18. | 

The cross-axis in the Fig. 26 is the line through {p, p') 
parallel to IJ' by the preceding Art. ^ 

57. In the following articles we will establish a property of 
homographic pencils which is similar to the cross-axis property 



J 



15-58] 



CROSS-CENTRE 



43 



cr( 

lb 



of homographic ranges proved in Art. 50, and may be called the 
cross-centre property of homographic pencils, viz. : 

Given three rays A, OB, OC of one pencil, and also three 
ys 0'A\ 0'B\ O'C of a second pencil, the line joining the 
points of intersection of OB, O'C and OC, O'B', and the line 
joinhig the points of intersection of OC, O'A' and OA, O'C are 
concurrent with the line joining the points of intersection of 
OA, O'B' and OB, 0'A\ 

To establish this we shall prove that these lines are each 
concurrent with the lines of the two pencils which correspond to 
O'O and 00' , so that the point of concurrency is a fixed point, 
which is called the cross-centre of the two pencils in analogy to 
the cross-axis of two ranges. 

58. Given two homographic pencils, vertices 0, 0', required to 
find 

(1) The ray in the second pencil corresponding to the ray 00' 
in the first ; 

B' A, 




Fig. 27. 



I 



44 CROSS-RATIO GEOMETRY [CH. V 

(2) The ray in the first pencil corresponding to the ray 00 in 
the second. 

We will employ the method of Art. 48. 

Let OA, OB, OC be three rays of the first pencil, and 0'A\ 
O'B', O'C their corresponding rays in the second. Let OA, O'A' 
meet in a, OB, O'B' in h, OC, O'C in c, OC, O'B' in S. Join ah, 
meeting OC in c, and 00' in I, and join ac meeting O'B' in h', and 
00' in m'. 

(1) Join SI meeting ac' in I'. Then O'V is the ray corre- 
sponding to 00'. For {ahcl) and {ab'c'l') are the ranges in which 
the two transversals ah, ac are cut by a pencil, centre S, and are 
therefore equicross by Art. 21. ^| 

(2) If ac meets 00' in m!, and Sm meets ah in m, then Om 
is the ray corresponding to O'O. 

Let the rays Om, O'V meet in T. Then T is, b> known point, 
and has the important properties given in the following Arts. ^ 

59. Given two homographic pencils {ABC ...) and 
0' {A'B'C ...), if any two non-corresponding rays OA, O'B' inter 
sect in 2, and the rays OB, O'A' intersect in 2', then 22' will 
pass through the fixed point T. 

For the pencils 0{ABTO') and O'(A'B'OT) are equicross. 
Therefore if we cut them respectively by the transversals O'A' 
and OA, the ranges (a^'u'O') and (a^Ou) are equicross, and 
since they have a common point a, they are in perspective by 
Art. 23. And since Ou, O'u intersect in T, the line joining 22' 
must also pass through the same point T. 

60. Def. The fixed point T may be called the homographic 

or cross-centre*. 
Cross-centre. 

Since OA, O'B' are any two non-corresponding 

rays, the position of T can be determined by means of the 

characteristics OA, OB, OC, and O'A', O'B', O'C. For if OB, 

* See note p. 40. 



i 

I 



i 



>9-61] CROSS-CENTRE 45 

'C intersect in S\ and OC, O'B' in S, the point of intersection 
)f *S>S" and 52^ is 2\ the cross-centre. 

61. The cross-centre enables us in theory, at any rate, to 
[solve the problem to construct a pencil homographic to a given 
pencil in a very simple manner. Thus, to find the ray corre- 
sponding to any given ray OM, let OM meet any ray O'A' in S. 
Join TS meeting OA in S'. Then O'S' is the ray required. 

The only difficulty with this method is that the cross-centre 
is often a distant point, or that the construction lines so fre- 
quently intersect at an inconvenient distance. Consequently 
the method of Art. 48 is in practice usually more convenient. 

If the cross-centre method is used, the best way to determine 
the position of 2^ is that given in Art. 60. 



CHAPTER VI 

METRICAL PROPERTIES OF HOMOGRAPHIC RANGES. THE 
CONSTANT OF CORRESPONDENCE. HOMOGRAPHIC EQUA- 
TIONS. ONE-TO-ONE CORRESPONDENCE 



62. If m, m! are any variable pair of corresponding points, 

{ahem) = {ab'c'jn). 

ac ^ be ac b'c 
am bm a'ni ' b'm ' 



am am, 
bm ' b'm' 



ac ac 
be ' b'c' 



Let us denote this compound ratio 7- : jy-j between segments 
given by the characteristics by the letter fx. 



Then 



bm 



= ix 



am 
6 W • 



in which /a is a constant, 



Therefore the equation -^ — — ix . -, , , 
bm bm 

represents two homographic ranges in which a, a and b, b' are 

two pairs of corresponding points. 

From this equation we can find any number of points m! on 
the second range corresponding to given points m on the first. 

Here the character of the homography is given either by 
(1) (a, a), (6, b') and a third pair of points (c, c) \ or by (2) (a, a'), 
(6, h') and the value of /x ; in fact, we only require to know the 



62-63] METRICAL PROPERTIES 47 

corresponding lengths ah, ah' on the two ranges, and the value 
of /x. 

Hence, ivhen two straight lines are divided homographically, 
the ratio of the distances of any point of division mfrojn two fixed 
points on the first is equal to the ratio of the distances of the 
corresponding point of division m from the two corresponding 
fixed points on the second range multiplied hy a constant. Con- 
versely, 

When two variable points m, m divide two fixed segments ah, 

_. _, . ^ J am am' , 

ah on two lines L, L %n such a manner that z — = ix . j-, — , , where 

hm '^ hm 

fx is a constant, L and L' are divided homographically hy the 
jyoints m, m. 

63. Given ahc, a'h'c the characteristics of two homographic 
ranges, the positions of /, J' can be obtained metrically by actual 
calculation of the lengths al, a J' as follows: 

(ahcl)^ (a'h'c cc'), 

ac be a'c' h'c a'c 
al ' hi a' GO ' * b'co ' h'c ' 

al ac a'c 
'''hi"h'c''l7^'^^' 

.'. al — fxhl —- fi(al - ah), 

.*. al (fx-l) -^ fxah, 

.'. al = . ah. 

fX-l 

In the same way it will be found that 

a'J' 1 , ,, ah' 

aJ = 



h'J' fx' l-fx 

al ah 



Hence hi = - = z , and h'J' = ixa'J' = . ah'. 

jX JJL — i I — u. 



48 CROSS-RATIO GEOMETRY [CH. v| 

64. If m, m' are a -pair of corresponding points of the range 
to prove that Im . J'w! is constant. 

Since (aI'moo) = (acc'm'J'), 

am a'm! 

.'. (Im- la) J' a + (J'W — J 'a') Im = 0, 
.'. Im . J'm'= la . J'a'= const. X (say). 

65. The converse of the property found in the preceding 
article is a very important one, and tells us that jH 

If on two given straight lines we take fixed points /, J', and^ 
also two variable points m, m! such that Im . J'rri is constant^ then 
the ranges (m) and {m) will he homographic, and the points /, J' 
will correspond to the points at infinity in the two ranges. 

The constant \ is called by Steiner " the power of the corre- 
spondence," and its actual value is seen from Art. 63 to be 
^ ac a'c' ^m 

It should be noticed that X is an absolute constant, which 
holds for all pairs of corresponding points on the ranges, whilst fx 
depends on the characteristics ahc, a'h'c', so that if we take two 
fixed points on the lines and call them / and J\ all that 
require to fix the homography is the value of X, whereas, if 
employ /a, we must know the values of ah, ah' and /a. 




Homographic Equations. 

66. The relation between any pair of corresponding points 
m, m' in two homographic ranges can, as we have shewn 
expressed by equations, of which the simplest form is 

I. Im.J'm' = Ia.J'a' ..(1) Art. 6- 

Here the origins are /, J'. 



64-66] HOMOGRAPHIC EQUATIONS 49 

If the ranges are on two separate lines, the points /, J' may- 
be at their point of intersection. If the ranges are co-axial, and 
/, J' coincide, we have the case of involution, Chap. IX. 

II. The next in order of simplicity is 

am am .„. . ^ co 

^— = ^. ^jj-, (2) Art. 62. 

bm ' hm ' 

Here there are two origins on each line, viz. a, h ; a, V. 

III. From (1) we can obtain other forms which we shall find 
useful, in which there is a single origin taken on each line, con- 
sisting of any chosen pair of non-corresponding points which we 
shall call a, V . The variable quantities in these equations will be 
am^ h'm\ where m, m! are a variable pair of corresponding points. 
If a, a' and 6, h' are pairs of corresponding points, writing 
Im = am — aly and J'm' = b'm — b'J', we have 

(am - al) [b'm' — b'J') = al . a' J', 
.'. am . b'm' - b'J'. am - al . b'm' + al. b'J' = al . a' J', 
and b'J' - a' J' = b'a', 

.'. am. b'm' -b'J' . am — al. b'm' + al . b'a'= (A). 

Again, since by Art. 64 

al _ bl _ al— bl _ ab 

V7'~'^'" yr^^^^u' " 6V ' 

therefore al . b'a! = b'J' . ab. 

Hence the relation (A) may be written 

am . b'jri' - b'J' . arn - al , b'm' + b'J' .ab = (B). 

From Art. 63, putting al = -^ . ab, b'J' = -^ . b'a' in (A) 

/*— 1 //,— 1 ^ ^ 

and (B), these become 

(1 - fi)am . b'm' + fx . b'a' .am + fi.ab. b'm' -fx.ab . b'a' = 0. . .(3), 
where jx is an absolute number which ilepends solely on the 
characteristics of the ranges, and can have any values except zero 
and infinity. 

M. 4 



50 CROSS-RATIO GEOMETRY [CH.^^B 

IV. If the origins coincide at the intersection of the 
ranges (when it is not a common point), i.e. when b' coincides 
with a, we have merely to write a for b' in (3) and we obtain j 

(1 — fi) am . am' + fiaa' . am + fxab . am' - /jcab . aa' = 0. . .(4), 

or am.am' — aJ'.a7n — aI.am' + aI.aa' = (4'). 

The ranges are not in perspective. 

Note. In equations (4) and (4') the quantities am', aa', aJ' 
could not exist unless a were on the line L', as all measurements 
are between points on the same line. Hence in these quantities 
a stands for the coincident point b'. 

In all the above equations, except (2) where four origins are 
used, the origins have been non-corresponding points. 

V. If the origins are required to be a pair of corresponding 
points a, a, we shall find it best to go back to (1), and write in 
it Im = am — al, J'm' = a'm,' — a' J', from which we shall obtain 

am . a'm' — a' J' . am — al. a'm' = (5'), 

or (1 — fi) am . am' — a'b' . am + fxab . a'm' = (5). 

The ranges may be, but are not necessarily, in perspective. 

YI. If the ranges are in perspective, so that their inter- 
section is a common j)oint, and if this point is taken as the ^ 
common origin a, a', then writing a for a' in (5) we have H 

(1 — /x) am . am! — ab' . am + fxab . am! = (6), ^| 

or am.am' — aJ' .am-al .am' =0 (6'), ^^ 

and the line mm! passes through a fixed point. S 

It is interesting to notice that the coordinates of this fixed 
point, referred to the two lines as axes, are (al, aJ'). For if 
(X, Y) is on mm', we have 

— + — 7 = 1, 

am am 
i.e. am . am! — Y . am ~ X . am' = 0, 

whence X=al, and Y = aJ'. 



66-67] HOMOGRAPHIC EQUATIONS 51 

This is, in fact, only a restatement of what we have already 
shewn geometrically in Art. 56. 

From V and VI we see that 

If the origins are a pair of corresponding points, the homo- 
graphic equation has no absolute term, and, conversely, if the 
hom,ographic equation has no absolute term, the origins are a pair 
of corresponding points. 

Also, when the intersection of the ranges is taken as the common 
origin, if the homographic equation has no absolute term, the 
ranges are in perspective. 

It only remains for us to consider the case when //, = 1. 



Proportional Section 



67. When /x= 1, the equation (2) of Art. QQ becomes 

am bin ab 
—, — ; = 7-7 — 7 = -TTi — const. 
am om, ab 

Thereforfe the lines are divided similarly, or into proportional 

parts. And since, by Art. 63, — - =^= 1, the point / must be 

a point at infinity on the line L. Similarly J' is a point at 
infinity on the line L'. So, conversely 

If tivo straight lines are divided into pi^oportional parts, they 
are divided homographically, and if the points /, J' are at infinity, 
the two ranges are similar. 

Of course, since / is by definition the correspondent of a 
point at infinity, the condition that / should be at infinity is 
equivalent to the condition that the ranges should have a pair of 
corresponding points at infinity, and so for J'. 

* On this form of homographic division Apollonius wrote his treatise 
de Sectione rationis in two books containing 181 propositions. This work, 
which was extant in Greek at the time of Pappus, was discovered in an 
Arabic MS and translated into Latin by Halley in 1706. See Art. 88. 

4—2 



52 CROSS-RATIO GEOMETRY [CH. VI 

68. If we put /A == 1 in the other equations in Art. 66, 

VII. (3) becomes -:r + TFT = 1 (7), 

^ ^ ah ha 

the origins being the non-corresponding points a, h'. 

VIII. (4) becomes ^+^'=.1 (8), 

^ ^ ah aa 

the origins being at the intersection of the ranges, which, ho"^ 

ever, is not a common point, and the ranges are not in perspective. 

IX. (5) becon.es ^=^ (9), 

^ ' ah ah 

the origins being the corresponding points a, a'. 

The ranges may be, but are not necessarily, in perspective. 

X. (6) becomes -—- = -— (10), 

the common origin being the intersection of the ranges, which is 
a common point, and the ranges are in perspective. 

If the lengths of the segments from m and m' to the origins are 
denoted by x, x, the equations in Arts. 66 and 68 are of the form 

Axx + Bx-\- Cx + B = 0, 
and may be divided into two classes. In the one in which the 
term xx' occurs the homography may be said to be of the second 
order, and in the other, where the term xx' is wanting, it may be 
said to be of the first order, the ranges being then divided 
similarly or proportionally. 

In homographic equations of the second order, 

( 1 ) If the origins are non-corresponding points 
neither A nor D can = 0. 

If (7=0, the origin for x is at 7. 

If -B = 0, the origin for x' is at J'. 

li B=0 and C = 0, the origins are at /, /'. 

(2) If the origins are corresponding points 
D = 0, but neither A, B, nor C can vanish, and the equati( 
must be of the form Axx + Bx + Cx' = 0. 



68-69] ONE-TO-ONE CORRESPONDENCE 53 

In homographic equations of the first order, 

(3) If the origins are non-corresponding 2)oints 

neither B, C, nor D can vanish (for then an origin would be at 
infinity), and the equation is of the form Bx + Cx' + D = 0. 

(4) If the origins are corresponding points 

D = Oy and the equation is of the form Bx + Cx' = 0, and we draw 
the same conclusions as in the case of homographic equations of 
the second order given at the end of Art. 66. 

One-to-One Correspondence. 

69. In both orders of equations corresponding to any given 
value of x there is one and only one value of x\ and correspond- 
ing to a given value of x there is one and only one value of x. 
When this is the case x and x' are said to be connected by a one- 
to-one or (1, 1) correspondence, and the ranges marked out on the 
two lines by giving different values to x or x' are homographic. 

It should be noticed, however, that for the equation 

Axx' + Bx + Cx' + D = 

to give two homographic ranges, we must not have A : B = C :D. 
For in that case the expression on the left hand could be 

factorised, and we should have (Ax + C) Ix + -jj^O, and there- 
fore one of the two variables must have a certain definite 
value, the other being then free to take ani/ value, i.e. in a 
factorising homography, all points of either line correspond to 
a single point of the other line *. 

The following examples are intended to illustrate the meaning 
and application of the homographic equations given in Arts. 66 — 
68. If the relation between a variable pair of points on two 
straight lines is of either of the types (6) or (10), their intersec- 

* For a geometrical illustration of this, see an article on **The double 
six " by G. T. Bennett, in Proceedings of the London Mathematical Society, 
p. 336, April 25th, 1911. 



54 CROSS-RATIO GEOMETRY [CH. VI 

tion being a common origin, then when x = 0, x' also -■ and the 
intersection of the ranges is a common point. The ranges are 
then in perspective, and the lines joining pairs of corresponding 
points pass through a fixed point. 

If the equation expressing the relation is of any of the types 
(3) to (10), and from the ranges pencils can be formed having 
the line joining the vertices for a common ray, the pencils are 
in perspective, and the locus of the intersections of pairs of 
corresponding rays is a straight line. 

It will be seen in Chap. XI that if the ranges are homographic 
but not in perspective the lines joining pairs of corresponding 
points will envelop a conic touching the ranges ; and if the 
pencils are not in perspective the locus of the intersections of 
pairs of corresponding rays is a conic passing through the centres 
of the pencils. 

In forming the homographic equation connecting two ranges 
we can generally determine its order by inspection from the 
consideration that if the points /, J' are at a finite distance the 
equation is of the second order, whilst it is of the first order if 
they are at infinity. 

EXAMPLES. 

1. A line through a fixed point P on the base BG of a triangle ABC cuts 

the sides AB, ^C in points m, m'. Find the homographic equation for the 

ranges (w) and {m') taking (1) A as common origin, (2) B and C as origins, 

and deduce from them the positions of the points I, J' on the sides ^B, AC. 

(1) Let Am = x, Am' = x'. Then by Menelaus' Theorem, Art. 18, 

Am'. CP.Bm=Cm' .BP.Am (A), 

.-. x' .CP.{x-c) = {x' -b).BP.x, 
.-. xx'{BP-CP)-b.BP.x + c.CP.x' = 0, 

.-. xx'--.BP.x + -.CP.x' = 0. 
a a 

Comparing this with the homographic equation (6') of Art. 66, VI, we 

& c 

deduce that ^ J' = -.£P; AI= -- .CP, as is of course obvious from the 
a a 

geometry of the figure. 



I 



EXAMPLES 



56 



(2) Let Bm=x, Gm' = x'. Substituting in (A) we have 
{h-\-x') CP .x = x' .BP{x + c), 
:. xx'{BP-GP)-b.GP.x + c.BP.x'=:0, 



XX'--. CP.X + -.BP.X' 
a a 



■0, 



by Art. 66, V, (5') CJ' 



:-.CP; BI-. 
a 



.BP. 



We might also have approached the question in the following manner. 

(1) The ranges (m) and (m') are obviously two homographic ranges of 
the second order and in perspective. Therefore by Art. 66, VI their homo- 
graphic equation is 

xx' - AJ' .x-AI. x' = 0, 

and from the geometry of the figure 

Ar = -.BP, AI=--.CP. 
a a 




Fig. 28. 



(2) When B and C are origins, by Art. 66, V, the homographic equation is 
xx' - CJ' . X - BI . x' = 0, 



and 



CJ' = -. CP, BI=-~,BP. 
a a 



2. Through the angle C of a parallelogram ABCD a straight line is 
drawn meeting the two sides AB, AD in a, a'. Prove that the rect. Ba . Da' 
is constant. 



56 CROSS-RATIO GEOMETRY [CH. VI 

The ranges (a), {a') are obviously homographic. When a' is at infinity, 
a is at B, and when a is at infinity, a' is at D. Therefore by Art. 64 
Ba . Da' is constant. 

3. AB, AC are two given straight lines of lengths a, h, in which points 
P and Q are taken such that AP :AB = AQ : QC. Prove that the straight 
line PQ passes through a fixed point. 

AP:AB=AQ:QC 

= AQ:AC-AQ, 
.: AP{AC-AQ) = AB.AQ, 
.'. x{b-x') = ax', where AP = x, AQ = x', 
.: xx' -bx + ax' = 0, 

.'. by Art. 66 (6) (P) and (Q) are homographic ranges, the origin A being a 
common point. They are therefore in perspective, and PQ passes through a 
fixed point. 

4. In Ex. 3 if D is the mid-point of AG, shew that the fixed point is the 
intersection of BD with the line through C parallel to AB. Prove that if E 
is taken on AB such that AE= -AB, the fixed point lies on the parallels to 
the given lines drawn through E and C. Shew also that E coincides with I 
and C with J'. 

5. Given the base AB of a triangle ABC, and the length of the segment 
mm' which the sides intercept on a straight line PQ parallel to AB, shew 
that the locus of the vertex C is a straight line. 

(m) and (m') are homographic ranges, being identical, and A (m), B (m') 
are homographic pencils. When m is at infinity, so also is vi'. Hence AB 
is a common ray, the pencils are in perspective, and the locus of C is a 
straight line. 

What is the force of the limitation that PQ is parallel to AB ? 

6. In Ex. 5 if a' and b are fixed points on PQ, and the ratio of the 
segments am, bm' is given, shew that the locus of the vertex is a straight 
line. 

Let am= k . bm'. Then by Art. 67, [m) and {m') are homographic ranges, &q. 

7. Q, B are fixed points in BC, the base of a triangle ABC. A line 
mm' parallel to the base meets the sides AB, AC in m, m'. Shew that the 
locus of the intersection of Qm, Em' is a straight line. 

Am AB 
/. by Art. 68 (10) the ranges (m) and (m') are homographic, &c. 



EXAMPLES 57 

In Ex. 7 if Cm, Bm' meet in P, shew that the locus of P is a straight 
fine. Shew that the same results hold in Exs. 7, 8 if vim', instead of being 
parallel to the base, cuts it in a fixed point. 

9. If on AB, the base of a triangle ABC, we take any length AT, and at 
the other end of the base another length BS in a fixed ratio to AT, and draw 
ET and FS parallel to a fixed line CR, meeting GA in E and CB in F, shew 
that the locus of 0, the intersection of EB and 2^^, is a straight line. (For 
analytical solution see Salmon's Conies, p. 45, Ex. 5.) 

By Art. 66 the pair of ranges (T) and [S) are homographic, as are also 
(T) and {E) and also [S) and {F). Therefore by Art. 39 the ranges {E) and 
{F) are homographic, and therefore also the pencils B {E) and A (F). 

Now when £ is at ^, T is at ^, S is at B, and F is at B. Hence AB is a 
common ray of the pencils which are thus in perspective and the locus of 
is a straight line. 

10. OA, OB are two given lines, m and m' a pair of corresponding 
points of two ranges on them whose homographic equation is of the first 
order. If the perpendiculars at m and m' meet in P, shew that the locus of 
P is a straight line. 

By Art. 68 (8) the ranges (m) and (m') are similar. The series of 
perpendiculars Pm constitute a pencil of parallel rays whose centre is at the 
point 00 , and the perpendiculars Pm' form a pencil whose centre is at oo '. 
The ranges (m) and {m') being homographic, so also are the pencils oo (P) 
and 00 '(P). Also, when m is at infinity, m' is at infinity, and the rays 
ccm, cc'm' coincide in the line at infinity. Therefore the pencils, having 
a common raij, are in perspective, and the locus of P is a straight line. 

11. Find the locus of the orthocentre of the triangle two of whose sides 
are given in position, and whose base passes through a fixed point. 

Let be the vertex of the triangle, P the fixed point, Pmm' any position of 
the base, H the orthocentre of the triangle 0mm'. Then the ranges (m), {m') 
are homographic, their equation being of the second order. The series of per- 
pendiculars mH, m'H form two homographic pencils, centres oo , oo '. The line 
joining their centres is the line at infinity, but this does not pass through m 
and m' at the same time, for m and m' are not at infinity together, as the 
ranges are not similar. The pencils are therefore not in perspective. 

It will be seen in Chap. XI that the locus of if is a hyperbola. 

12. Two sides of a triangle are given in position, and their sum is 
constant. Prove that the centre of the nine-points circle traces out a 
straight line. 

Let AOB be any position of the triangle, and let OA = x, OB = x', so that 
X + x' = const. Let D be the mid-point of OA, BD' perpendicular to OA , and 



^H. vW 



58 CROSS-RATIO GEOMETRY [CH 

m the mid-point of BD'. Forming m' in the same -way, the perpendicular to 
OA through m meets the perpendicular to OB through m' in the nine-points 
centre. 

Then 20m:=^ + x' co^O, 

x' 
20m' = a:cos 0-f '— , 

.-. Om + Om' = 1(4 + cos 0) {x + x') = const. , 
.■. by Ex. 10 the locus of the nine-points centre is a straight line. 

13. Find the locus of the centre of the circum-circle of a triangh 
the position and the sum or difference of two of the sides are given. 

14. Given the position and the sum or difference of the reciprocals of 
two sides of a triangle, shew that the base will always pass through a fixed 
point. 

15. In Ex. 14, if Om, Om' are the sides, shew that the base will pass 
through a fixed point if the sides are connected by the relation 

Jc_ I 

Om Om'~ ' 
where k, I, n are constants. 

16. OA, OB are two given straight lines, A and B fixed points, 
points P on OA and Q on OB vary in such a manner that 

1 1 _ J^ 1^ 

OA 0P~ OB OQ' 

shew that PQ passes through a fixed point. 

17. OA and OB are two given straight lines, and from a fixed pom 
two straight lines CM, CN are drawn to them so that the triangles OMN, 
CMN are equal. Shew that 3IN passes through a fixed point. ^H 

18. OP, OQ are fixed lines, and the circum-centre of the triangle 0P(^^ 
lies on another fixed line. Shew that P and Q are corresponding points of 
two ranges of the first order not in perspective. ^m 

19. Through a fixed point two straight lines OPQ and OP'Q' are drawn^ 
meeting two fixed parallel straight lines. If PQ' and P'Q meet in R, prove 
that the locus of ii is a straight line. 

20. In Ex. 19 if the two fixed lines are not parallel, shew that the 
of JS is a straight line. 




EXAMPLES 59 

21. Through a fixed point a straight line is drawn meeting two fixed 
parallel straight lines in P and Q respectively, and through P and Q straight 
lines are drawn in given directions intersecting in R. Prove that the locus 
of JR is a straight line, 

22. ABC is a triangle, MN is any straight line parallel to AG^ cutting 
the sides BC, BA of the triangle in M and N respectively. Shew that the 
locus of the intersection of AM and GN is a straight line. 

23. A and B are fixed points in a line, and. C, D are fixed points in 
another line parallel to AB. Find the locus of a point P such that if PA, 
PB meet GB in Q and J?, the sum of GQ and DR is constant. 

24. L, L' are two fixed lines, ABC a triangle whose base BC, of 
constant length, slides along L, and the vertex A moves along L' in such a 
way that AB is always parallel to a given direction. If the side ^C is 
divided in a constant ratio at K, the locus of ^ is a straight line. Use 
Chap. IV, Ex. 1. 

25. L, L' are two fixed lines, m, m' are a pair of corresponding points 
on them in the two ranges given by the equation Bx + Gx' + Z) = 0. If mm' is 
joined and divided in a constant ratio at h, the locus of ^ is a straight line. 
Newton, Princip. Bk i, Lemma xxiii. For the case v/here the equation 
between m, m' is of the second order, see Chap. XI, Ex. 13. 

26. Given in magnitude and position the vertical angle of a triangle, 
and the sum or difference of the sides containing it, the locus of the mid- 
point of the base is a straight line. 

27. A parallelogram is inscribed in a triangle, having one side on the 
base of the triangle, and the two sides adjacent to it parallel to a fixed 
direction. Prove that the locus of the intersection of the diagonals of the 
parallelogram is a straight line bisecting the base of the triangle. 

28. OA, OB are two given straight lines. The points P on OA and Q on 
OB vary in such a manner that the ratio of ^iP to ^Q is constant. Shew 
that the locus of the mid-point of PQ is a straight line. 

29. OA, OB are two given lines, m, m' a pair of points on them such 
that the perpendiculars to OA at m and to OB at m' meet on a fixed straight 
line. If through in, m' are drawn parallels to the given lines, shew that the 
locus of their intersection is a straight line. 

30. OA, OB are two fixed lines, m, m' a pair of corresponding points on 

k I 

them in the ranges given by the relation -:— + pr—, = n, where Jc, I, n are con- 

Om Om 
stants. If through m, m' are drawn parallels to the given lines, discuss the 
question whether the locus of their intersection is, or is not, a straight line. 



CHAPTER VII 

TWO HOMOGRAPHIC CO- AXIAL RANGES. THEIR COMMON 
POINTS, AND METHODS OF FINDING THEM 

70. Hitherto we have supposed the two homographic ranges 
to be on two separate Hnes L and L'^ on each of which are given 
the three arbitrary points a, 6, c and a\ h', c. Now if m is any point 
of the range on Z, the corresponding point m' on L' is given by 
the relation {m'a'h'c') = (mabc), which shews that the position of 
m on L' is quite independent of the angle between L and L'. 
Hence in Fig. 24, p. 39, if we suppose L' to rotate about q' until 
it coincides with L, each of the points of the range on L' will 
describe a circle with q' as centre, and will still be at the same 
distance from p (or q^) as before, the homography of the ranges 
will be unaltered, and p being now the common origin, the 
relation (4') of Art. 66 representing two homographic ranges 
on the same straight line becomes 

pm . pm' — pJ\ pm —pl.pm' +pT.pp' = 0. 



Common Points. 



i 



71. When the ranges are on different lines their point of 
intersection might or might not be a common point 
of the ranges, a common point being defined as a 
point of coincidence of two corresponding points of 

the ranges, and of course there could not be more than the one 

common point. 



J 



i 



-73] COMMON POINTS 61 



When, however, the ranges are co-axial there will usually be 
two common points, but never more than two, unless the ranges 
coincide. These common points are of great interest and import- 
ance as will be shewn by examples in the following chapter. 

They are obtained by putting m for m' in the relation of 
Art. 70, which gives us 

pm^ - (pi + pJ') pm + pi . pp' — 0. 

This being a quadratic equation gives us two and only two values 
of pm, and these of course may be real and unequal, coincident, 
or imaginary. 

If e and / are the points given by these values of pm, they 
are the common points of the ranges. Some of their properties 
are given in the following articles. 

72. The mid-point of ef is also the mid-point of IJ' . 

For if is the mid-point of ef by the equation in Art. 71 we 
have ^pO=pI-vpJ'. 

The coincidence of the mid-points may also be proved by using 
Art. 64. 

For since X = el . eJ' = If. Jf 

. e/ _ //•_ el+If ef 
' ' J'f~ eJ' ~ eJ'-^Jf Vf ' 

.'. eI = Jf, and If^eJ'. 

Therefore the mid-point of e/is also the mid-point of IJ'. 

73. If 0, the mid-point of ef is a point on the first range, 
and 0' its correspondent y the range {efO'J') is harmonic. 

Transferring the origin from p to 0, in which case pO is zero, 
we have 

Oe''+OI.OO' = 0, 

and since 01 = - OJ', and Oe^ = Of^, this becomes 

Oe''=Of'=OJ'.00' (A), 




J 



62 CROSS-RATIO GEOMETRY 

and therefore by Art. 32 {efO'J') is a harmonic range. 

follows from (A) that ^H 

The common points are real or imaginary according as OJ' atic^ 
00' have the same or opposite signs, i.e. according as J' and 0' are 
on the same or opposite sides of 0. ^H 

The above relation (A) can also be obtained by using the 
constant of correspondence, Art. 64. 

Thus, from Im . J'm' - const. = la . J 'a!, 
we have {Om - 01) {Om' + 01) = {Oa - 01) {Oa! + 0I\ 
and therefore Om . Om' - mm! . OI=Oa. Oa' - aa . 01, 
as the general relation connecting the distances from of 
corresponding points. In particular, for common points this 
becomes J| 

Oe^^Oa.Oa'-aa'.OI, ^ 

shewing at once that there are two common points equidistant 
from 0. If a coincides with 0, 

Om . Om' - mm' .01 = -00'. 01= 00' . OJ', 
.-. Oe' = 0/' = - 00' . 01 = 00' . OJ'. 

The cross-ratio formed by the common points and 
any pair of corresponding points is constant. 

74. Taking the relation given in Arts. 62, 66 
am a!m' 
hm h'm ' 
since a, a' and 6, h' may be any two pairs of corresponding points, 
let a be the common point e, and b the common point f. Then 
a coincides with e, and h' with/, and the relation becomes 

e7n em! 



where ju, is const. 



fm ^fm" 

em em' , ^ ,. 
f^ = -^— : -T—, = (ejmm ). 
fm Jm ^^ ' 



'3-78] {aa'ef) is constant 63 

75. The above result may also be obtained as follows. 
Since {abef) = {a'h'ef), 

, ae he _ a'e h'e 

''af'-hr-^f'-hT 

. cLe a'e he h'e 

"af''Vf=bf''¥f' 

:. {aa'ef) = (hh'ef). 

Hence the cross-ratio of the range formed hy the common points 

and any pair of corresponding points is constant. 

76. There cannot he more than two com,mon points. 
For suppose there were three, viz. e^f g. Then 

(efgm) = {efgm'), 

therefore m' would always coincide with m, and the two ranges 
would be identical. See Art. 71. 

77. If one of the common points, as /, is at infinity, since 

{meaco ) = (mWoo ), 
. ma _ m'a' 

ea ea' ' 

. ae _ am ah 

a'e am! a'b' ' 

Therefore the lines are divided proportionally. 

Conversely J if two ranges are similar, one of the common 2)oints 
is at infinity. See also Art. 67. 

If in addition the ranges are equal and in the same sense, so 
that ah = ah', then ae =■ a'e, and therefore e is at infinity, i.e. if 
the ranges are superposable, both common points are at infinity. 

78. By Art. 73 Oe^ = QJ' . 00'. 

If the point 0' coincides with 0, we see that Oe vanishes, 
the common points coincide at 0, and the constant of correspon- 
dence = Im . J'm' = 10 . J'O = - \IJ''^. 






64 CROSS-RATIO GEOMETRY [CH 

The case where coincides with J' belongs to the system in 
involution, and will be considered in Chapter IX. 00' is then 
infinite, whilst the product OJ' . 00' is still finite. 

By Art. 64 \ = Ie.Je 

= {0e-0I){0e +01) 

= Oe^-OI\ 

.'. Oe' = OP + \=Of\ 

From this we see that if X is positive, Oe is always real. 
If A, is negative, and < OP^ Oe is always real. 
If X is negative and = — OP, Oe vanishes, and the comi 
points coincide at 0. 

If \ is negative and > OP, the common points are imaginary. 

d' e c' J' f O' b' 



h 


d 1 


ab 6 

Fig. 29. 






L 






d' eO' 


J' 

1 


c' 


b' a' L' 


c 


ii i 


dfo 

Fig. 30. 






L 


c' 


O' 


d' 


J' 




a' bV 


a 


h 


i c 6 

Fig. 31. 






d L 



The dififerent cases of this article are illustrated by the above 
figures, each of which gives two co-axial homographic ranges. 
The divisions of L are shewn below the line, and those of L' 
above it. In Fig. 29 the common points are real and separate; 
in Fig, 30 they are real and coincident, in which case all the 
points e, f, 0, 0' coincide. In Fig. 31 they are imaginary, although 
their mid-point is real. 



i 



78-80] 



TWO CO-AXIAL KANGES 



65 



79. If we have two homographic ranges on the same axis, 
and if a is a point on the axis which has al or a!' for its corre- 
spondent according as we consider it to be a point in the first 
or second range, then, as the point a varies, the range which it 
describes will be homographic with the range (a"); and it is 
by supposition homographic with the range {a!). Therefore by 
Art. 39, the ranges {a!) and {a!') are homographic. Also, if a coin- 
cides with either of the common points of the ranges (a) and {a!\ 
the points cb and a" wall also coincide with it. Hence the ranges 
(a') and (a") will have the same common points as the ranges 
{a) and {a!). 

V 00' 




80. If two homographic pencils, vertices F, F', are cut by 
a transversal, we shall obtain two homographic ranges upon it, 
and if e, / are their common points, Fe, F'e and Yf, Y'f are in 
general the only pairs of corresponding rays which intersect on 
the transversal. If there are more than two such pairs, the 
pencils are in perspective, having the transversal for axis of 
perspective. See Art. 45. 

If we have two co-axial homographic ranges, and we join 
the points of division to an external point F', we shall obtain 
two homographic concentric pencils in which the rays drawn to 
the common points will be common raysy and the pencils will 

M. 5 



66 CROSS-RATIO GEOMETRY [cH. V] 

have properties similar to those possessed by two co-axial ranges 
of which the two most important are 

(1) hy Art. 74 the cross-ratio formed by the two commc 
rays and any pair of corresponding rays is constant^ 

(2) hy Art. 76 there cannot he more than two common rayi 

81. Given two homographic ranges on the same straight lit 
ij the common points are imaginary there are two points on^ 
opposite sides of the line at which pairs of corresponding jjoints 
suhtend equal angles. 

R 




J' O I C 

Fig. 33. 

Since, by Art. 73, 0' and J' are on opposite sides of 0, there- 
fore 0' and / are on the same side of 0, and the product 01. 00' 
is positive. Draw OR perpendicular to the given line of such 
length that OR^ = 01 . 00\ and through R draw a line parallel 
to the given line. '^^ 

Then J'RO' is a right angle, and the right-angled triangles 
ORI, 00' R are similar. Therefore the angle 

ORI=00'R^O'Rcx^\ 

the angle IR ao = oo 'RJ', and the angle OR oo = O'RJ'. 

Therefore in the two pencils R {01 oo ), R{0' co 'J') the a 
between each pair of rays of the one is equal to the angle between 
the corresponding rays of the other taken in the same sense. 
Consequently by Art. 42, if m, m' are any pair of corresponding 
points in the given ranges, the pencils R (OIco m), R (0' co 'J'm') 
are superposable by rotation through the angle ORO'. 

The point R\ the image of R on the other side of O'J', will 
course have similar properties. 



angl^^ 



I 



81-82] ROTATION CENTRES AND ANGLE 67 

Hence, if either of these points is found, and also any pair 
of corresponding points (m, m'), any additional number of pairs 
can be found by means of the constancy of the angle mRm. 

Dep. We might call these points i?, R' the rotation centres 
Rotation cen- ^^ *^® homography, and the constant angle between 
tres, Rotation corresponding rays the rotation angle. 
^^S"^^- These points can be obtained without finding 

and 0' if we know the power of the correspondence A., i.e. if 
we know /, /' and any pair of corresponding points (m, m). 
For OR^=OI.OO\ 

:. RP = OR^ + OP = 01 (00' + 01) = 01 {00' - OJ') = 01 . J'O'. 
:. RP = RJ'^= -\. 

Therefore RI and RJ' are equal to the mean proportional 
between ml and J'm!. 

Another method of obtaining the positions of the rotation 
centres is by means of the circle in Fig. 36, Art. 84, by finding 
the tangential distance of I or J' from that circle, and drawing 
arcs with / and J' as centres and this tangential distance as 
radius. The points of intersection of these arcs will be the 
rotation centres, for (the tangent)^ = /Z> . J'D' = -X. 

82. We will now consider the question 

To construct on a straight line a row which shall be homo- 
graphic to a row already described upon it. 

We will give two methods of construction, and it will make 
the subject clearer if, instead of a single line, we imagine a 
double line consisting of two parallel lines Z, L' indefinitely near 
to one another. 

(1) Let abc be the characteristic of Z, and take any three 
points a, b', c to be the characteristic of L'. Art. 38. 

g' a' b' c l! 

J a I c L 

Fig. 34. 

5—2 



CROSS-RATIO GEOMETRY [CH. 



Take any point on the double line and let it be denoted 
J) when considered as a point on Z, and by 5^' when considered 
as a point on Z', and rotate the line L' about c^ so as to make any 
angle with Z, the points a', h\ c retaining their distances from q' 
(or p) unchanged. As we remarked in Art. 70, since 

(abcm) = {a'h'cm), 
the distances of corresponding points m, m! from p are inde- 
pendent of the angle between Z and Z'. 

Then in Fig. 24, p. 39, by joining pairs of non-corresponding 
points (ab')j (a'b) and (ac'), (a'c), and drawing the line through 
their points of intersection y, j8, as in Art. 50, we obtain the 
cross-axis p'q, and the points (p, p') correspond, as do also 
{q, q'). By means of the cross-axis any number of pairs of 
points (m, m!) can be found. 

To find I and J\ and 0'. 

In Fig. 25, p. 41, let c, c' be any pair of corresponding points. 
Through c draw a line parallel to Z' meeting p'q in i. Then c'i 
produced will meet Z in the required point /. Through / draw 
a line parallel to p'q. This will meet Z' in the required point J' 
by Art. 55. 

Rotating L' about q' to its original position in the double 
line and bisecting Z/', we obtain the point 0. To find 0' let Oc' 
meet the cross-axis in w. Then cm will meet L' in the required 
point 0'. 

(2) Move one of the given lines Z, Z' parallel to itself until any 
pair of corresponding points, say (jo, p'), coincide, and rotate the 
line L' through any angle about this common point, as in Fig. 26. 
By Art. 41 the ranges are now in perspective, centre S, and if 
we take any point m on Z, and join Sm, it will cut L' in m\ the 
point corresponding to m. Draw SI, SJ' parallels to Z' and Z, 
and rotate Z' back again until it coincides with Z, and then move 
it parallel to itself until p' is in its original position. We shall 
then have two homographic ranges on the given line, with the 



YII ^H 

^by ■ 



1 



tne H 



82-84] 



CHASLES METHOD 



69 



points /, /' marked on it. It will be noticed that we have given 
two alternative methods of finding / and J\ but in most problems 
the conditions will enable us to determine these points without 
having to use either of the methods, and then the easiest way of 
finding further pairs of points will be the method of Art. 84, as 
it involves no moving of the lines. 



Methods of finding the common points e and f. 

83. Since Oe^= OJ' . 00\ Art. 73, Oe is a mean proportional 
between the two known lengths OJ' and 00\ and can therefore 
at once be found by Euc. vi, 13. 

We shall refer to this as Chasles' method of finding the 
common points (1852)*, 

84. The following simple method of finding the common 
points and constructing any number of pairs of corresponding 
points on the two ranges is due to Prof. Alfred Lodge (1907) f. 
It requires the finding of / and J\ but does not make use of 
and 0'. 




Fig. 35. 

* Traite de GeomStrie Superieure, Art. 154. 
t Mathematical Gazette, April, 1909. 



70 



CROSS-RATIO GEOMETRY 



[CH. VII 





,--''' 

,,'- 

y' 


1 


\ 


\ 


\ 






/ 


7K 


\^ 


\ 




/ 




\ 


\\ 


] 


' 


/ 




\ 


^^oy 






/ 








^ 


v\ 




I 




v^ 


p 


m 




^ 


~ 


P' 


m' 



Fig. 36. 
I'p . J'p' = constant of correspondence 
= Ie.J'e 
= Ie(Ie-IJ'), 
:. Ie^-IJ'.Ie + Ip.p'J' = 0. 

To solve this quadratic equation, graphically, measure from 
/ and J' vertical distances IB = Ip, and J'D' = p'J', ID and J'D' 
being drawn on the same or opposite sides of IJ' according as 
Ip and p'J' are measured in the same or opposite directions. 
Then the circle described on DB' as diameter will cut the given 
axis in the required common points e, f. 

For if ID (produced if necessary) cuts the circle again in Z>i, 

le . J'e = er.If= DI . IB^ = BI .J'B' = Ip . J'p'. 

The reason for putting the equation in the form 

le'-IJ' .Ie + Ip.p'J' = 

is to make it clear how the coefficients in any such quadratic 
equation are connected with the lines in the graph. The lengths 
and signs of the perpendiculars IB, J'B' are the equivalents 
of the factors of the constant term, and the distance between 
them measured from the origin / is equal to the coefficient of le 



84] lodge's method 71 

with its sign changed, so that in Fig. 35, If is the positive root 
and le the negative root, if we consider IJ' as positive. 

In Fig. 36, the roots are imaginary. In a vector sense they 
may be considered as represented by the rotation centres defined 
in Art. 81, since 0R^ = — Oe^, and the roots are 10 ±0e^ that is, 
JO ± OR J'^. 

We shall refer to the above as Lodge's method. 

Figs. 35 and 36 also give us a simple method of constructing 
any number of pairs of corresponding points on the two ranges. 

For let m be any given point on the first range, and let Dm 
cut the circle in tt. Join D'tt meeting the axis in m'. Then the 
triangles Dim, in'J'D' are similar. 

.-. Im.ID^J'D' :J'm\ 
:. Im, . JW = ID . J'D' = Ip . Jy. 

Therefore by Art. 65 the range m' is homographic to the 
range m. In Fig. 35 the common points are real, and in Fig. 36 
they are imaginary. If the ranges are such that the circle on DD' 
touches the axis, the common points will coincide at the point of 
contact, as will also the points and 0'. 

If we construct the Lodge circle for each pair of correspond- 
ing points we shall obtain a co-axial system since each passes 
through the points e, f. If the common points are imaginary, the 
circles have real limiting points which are the rotation centres, 
for the distance of each from the axis is equal to the tangential 
distance of from any circle of the system. 

Ex. \i A^B are two fixed points on a circle, and P a variable point on 
it, and if PA, PB produced cut a fixed line (which does not cut the circle in 
real points) in M and If', shew that MM' subtends a constant angle at R, 
where jR is either of the limiting points of the system of co-axial circles 
defined by the given circle and the given line as radical axis. 

Find I on MM' such that the angle AIM=APB, i.e. so that A, P, M% I 
are concyclic. Then I is a fixed point, and 

MI . MM' = MA . MP= (tangent)2=il/iJ2. 

Therefore the triangles MBM\ MIR are similar, and the angle 
MRM'^MIR, 
which is fixed. Therefore the angle MRM' is constant. 



72 CROSS-RATIO GEOMETRY [CH. VII 

85. In comparing the methods of Arts. 83 and 84 it will be 
noticed that they both depend upon first finding / and J\ which 
it will be found in practice can generally be determined by 
inspection from the conditions of a problem. We thus obtain 
the characteristics alcc and a'oo 'J'. Chasles' method then pro- 
ceeds to find 0\ and the construction is completed byEuc. vi, 13. 
By Lodge's method it is not necessary to find 0', and the 
amount of construction required is distinctly less. Moreover 
Lodge's circle, besides giving the common points, enables us to 
construct as many pairs of corresponding points as we please. 

86. The following method given by Chasles* enables us to 
find the common points directly from the characteristics ahc 
and a'h'c\ without finding / and J'. 

Through any arbitrary point g describe two circles having 
ah' and ha as chords, and intersecting in a second point g'. 
Through g describe two other circles having ac', ca! as chords, 
and intersecting again in g". Then the circle round gg'g" will 
intersect the given line in the common points required. 

We may remark that this method, which depends on the 
construction of five circles, is one which it is not easy to use, as 
it is difficult in practice to construct the circles with sufficient 
accuracy to obtain more than approximate positions of the common 
points. 

87. Another construction by means of a circle or conic will 
be given in Art. 160, at the end of Chapter XII. 

Note. Art. 84 may be treated analytically as follows : 
If the homographic equation of the two ranges is known to 
be of the second order, it can always be put in the form 

(a?i - a) (a?2 - 6) + lik = 0, 

where hk is never zero, i.e., in the form 

x^-a x^—o 
* Traite de Geom. Sup., Art. 263. 



1 



85-87] COMMON POINTS 73 

In Figs. 35 and 36, taking the common axis as the axis of x, 
suppose the point m is given (x^, 0). Find two points D, D' 
with coordinates {a, h) and (6, k) respectively, and describe a 
circle on DD' as diameter. From w, (ajj, 0), draw mD meeting 
the circle again in tt. Then ttD' will cut the common axis in 
m, (.x\^, 0), for 

the gradient of mD is , 

CL X-y 

k 
and the gradient of m'D' is j ;- , 

and by the homographic equation their product = — 1 . Therefore 
mD and m!D' meet on the circle. 

CoR. 1. It follows that the common points are the points 
where the circle cuts the common axis. 

Cor. 2. The point / is at (a, 0) and J' at {b, 0). 
Cor. 3. The equation of the circle is 

{x-a){x-h)^(y-h){y-k) = 0. 

Therefore when 2/ = 0, {x - a) {x — h) + hk = 0, giving the 
common points. 



CHAPTER VIII 

PROBLEMS OF THE THREE SECTIONS. — OTHER PROBLEMS 
WHOSE SOLUTIONS DEPEND ON FINDING THE COMMON 
POINTS OF TWO CO-AXIAL HOMOGRAPHIC RANGES j 

88. The problem of finding the common points of two homo- 
graphic ranges on the same axis is one of frequent occurrence, 
and can be applied to the solution of geometrical questions which 
in analysis would depend upon the solution of equations of the 
second degree, and we will therefore solve somewhat fully a few 
problems, which will enable the student to become familiar with 
the method. We will follow Chasles^ in selecting for our purpose 
three of the most noted problems of the ancients, viz. : 

(1) On Determinate Section, rys htoipLo-ixivq'; TOfx-fjs. 

(2) On Spatial Section, rrjq aTroro/x^s tov ^wptov. 

(3) On Proportional Section, riys aVoro/x^s tov Koyov. 
These problems are of interest in themselves, for "the great 

geometer " Apollonius wrote separate treatises on them, intending 
them to be text-books on the application of analysis to geometry. 
They were extant in Greek in the time of Pappus, 400 a.d., and 
the fertility of the problems and the thoroughness with which 
they were treated may be inferred from his statement that the 
first treatise contained 83 propositions, the second 124, and the 
third 181. It was generally supposed that they had perished, 



)ose 

J 



Traite de Geometrie Superieure, Chaps, xiv, xv. 



i 



88] TREATISES OF APOLLONIUS 75 

either from the destroying hand of time, edax rerum, or from the 
still more destructive hands of barbarians, until Edward Bernard, 
who was Savilian Professor of Astronomy from 1673 to 1691, 
discovered an Arabic manuscript containing the section of ratio 
in the Bodleian Library. Being skilled in Oriental languages he 
began to translate it into Latin, but at his death his successor, 
D. Gregory, found that he had hardly completed a tenth part of 
it, and at the suggestion of Aldrich, Dean of Christchurch, the 
manuscript was submitted to his friend Halley, who had just been 
elected Savilian Professor of Geometry, and who took a keen 
interest in the editing of ancient mathematical writers. Undis- 
mayed by the fact that he did not know a word of Arabic, he 
cheerfully undertook to complete Bernard's unfinished task, and 
in the preface to the work he tells us how he did it. With the 
aid of that part of the manuscript which Bernard had translated 
(which consisted of only 13 pages out of 138), he first picked out 
those words whose meaning he was able to recognise from the 
context, and then by studying the argument and turning over 
and over in his mind what might be the meaning of the words 
which he did not recognise, by this method of deciphering he 
groped his way through nearly the whole of the book, and 
obtained a general idea of its contents. Then by recommencing 
and going over the same ground step by step again and again he 
managed to complete the work without the assistance of anyone 
else. 

Having overcome this obstacle so successfully, it sounds 
almost hypercritical when he tells us that in addition to his 
other difficulties the manuscript was badly written, the diacritical 
points were wanting from many of the letters, and occasionally 
words, and even sentences, were missing, so that, as he says, it 
required a soothsayer rather than an interpreter to divine the 
true meaning. He then proceeded to restore the treatise de 
Sectione Spatii, and here he had not even the help of an Arabic 
version, but merely a short description of its contents given by 



[^^F 



76 CROSS-RATIO GEOMETRY [CH. 

Pappus in the preface to his seventh book of the Col. Math. 
together with a few Lemmas dealing with the subject. The two 
treatises were published by him in Latin in one volume in 1706. 

The treatise on determinate section was restored from similar 
scanty materials by Snell in 1601, by Lawson in 1772, by Wales 
in 1772, and by R. Simson in his posthumous works published in 
1776. 

The road which all these writers have taken is, however, a 
long and toilsome one, and as Chasles points out, homography 
gives us a simple method by which we can solve the problems 
either in their most general form, or in any of the particular cases 
which they can assume. 



I. Determinate Section. 



i 



89. Given four collinear points a, a\ b, h', it is required to 
find another j)oint m, collinear with them, such that 



ma . mh , , ^ PQ 

— 5 , ~ a (const.) = -^ 

mo . ma qr 



4 



There will evidently be two points satisfying the given con- 
ditions, for the equation is a quadratic, and these two points are 
obviously the common points of the two homographic divisions 

formed by — ^ == u . -^77 , where a, b are two points in the first 
-^ mb ^ 7nb 

row, and a', b' their correspondents in the second. See Art. 
To determine the points I and J'. 

Let m' be at 00 '. Then ^, = «,. 
lb 

Let m be at 00 . Then y,—, = u. 
J a 

Having found / and J', we can determine the common poini 
e, / by any of the methods given above. 



d 



89] DETERMINATE SECTION 

Chasles' method, Fig. 37. 



77 




J' 6 f b' 

Fig. 37. 

Bisect IJ' in 0, and considering as a point in the first 
row, find its corresponding point 0' from either of the relations 
aO a'O' 



hO 



h'O' 



or IO.J'0'=^Ia.J'a\ 



On 10' describe a semi-circle, in which draw OC perpendicular 
to 10'. Then the circle with centre and radius OC will cut the 
axis in the required points e,/. 

Lodgers method^ Fig. 38. 




a' 



le . J'e —la .J 'a', 
.'. Ie{Ie-IJ')+Ia.a'J'=(), 
:. Ie''-IJ' .Ie + Ia.aJ' = 0, 



78 CROSS-RATIO GEOMETRY [CH. VI 



On perpendiculars through /, J' take ID = la, and J'B' = a' J', 
measuring them on opposite sides of IJ', because la and a' J' are 
of opposite signs. Then the circle described on DD' as diameter 
will cut the given line in the required common points e, f. 



\ 




II. Spatial Section. 

90. Given two straight lines AL, BL' on which A and 
fixed points, it is required to draw through a fixed point P a trans- 
versal ePe forming on AL, BL' the two segments Ae, Be such that 
A e . Be = a given quantity K. 

Let a, a' be two points on AL, BL' such that Aa . Ba! = K. 
Then in their different positions the divisions {a) and (a) are 
homographic by Art. 65; and if a'P meets AL in a, the ranges 
{a) and (a) are homographic by Art. 39, and we have to find the 
common points of these ranges. 

When a is at infinity, a is at B, and a at J', the point wheri 
BP meets AL. 

When a is at infinity, a is at /', where PI' is parallel to AL^ 
and / is obtained from the relation AI . BI' — K. 

[The position of / can also be found as follows. If in Fig. 4' 
we rotate the line L' about /' so as to come into the position 
I'P, and if B, a, ... come into the positions B^, %', ..., then AB^ 
is the cross-axis of the ranges A, a, ..., B^, a{, .... There- 
fore if we join Fa cutting AB^ in X, a^X will pass through /j 
Similarly we can find O' in Fig. 39.] 

Employing Chasles' method. Fig. 39, find the mid-point of 
IJ', and Q! its corresponding point on BL' from the relation 
AO . BQ' = K, and join Q'P meeting AL in 0'. Describe the semi- 
circle on 10' and obtain the common points e, f as in the preceding 
problem. 

Join eP,fP and produce them to meet BL' in e',f'. Then, 
the two pairs of points e, e' and/,/' are the points required. 




90] 



SPATIAL SECTION 



79 




Fig. 40. 



Employing Lodge's method, Fig. 40, as in Art. 89, we have 
/e2_/J'.7e + /a.a'J' = 0. 



80 



CROSS-RATIO GEOMETRY 



[CH. VIII 



Find the points D, D\ which will be on the same side of IJ\ 
because la and a J', the factors of the last term, are both 
measured in the same direction. Then the circle on DD' as 
diameter will cut ^Z in the points e, y, and the problem can be 
completed as before. 



III. Proportional Section. 



A 



91. Given two straight lines AL, BL' on which A, B are fixed 
points, it is required to draw through a fixed point P a transversal 
ePe' forming on AL, BL' the two segments Ae, Be' which shall he in 
a given ratio \. 




Let a, a' be two points on AL, BL' such that ^^, = X. 



Then 



in their different positions the divisions (a) and (a') are homo- 
graphic by Art. 67; and if a'P meets AL in a, the ranges (a) and 
(a) are also homographic, and their common points will give us 
the points required. 



i 



91-92] 



PROPORTIONAL SECTION 



81 



To find /, the position of a when a' is at infinity, draw PI' 

AI 

parallel to A Z, and take -^j, = A, for a' then coincides with /'. 

To find J', the position of a when a is at infinity, and when 
a' is also at infinity by Art. 67, draw FJ' parallel to BL'. 

If we wish to use Chasles' method, find the mid-point of 
IJ'. On BL' find Q,' corresponding to from the relation 

AO 

-o?y = K ^^^ join O'P meeting AL in 0', and complete the con- 
struction as in the problems of Arts. 89, 90. 
Employing Lodge's method we have 

M-IJ' .le + Ia. a'J' = 0. 

The points D, D' must now be taken on the same side of IJ\ 
since la and a J' have the same sign. 

It has been left to the student to draw the circles according 
as he employs Chasles' or Lodge's method. 

92. Given a triangle ABC, and three points P, Q, R in its 
plane, it is required to inscribe in ABC another triangle whose 
sides shall pass through P, Q, R. 




82 CROSS-RATIO GEOMETRY [CH. VIII 

Through P draw any straight line PI a, cutting BC in a and^ 
AB in 1. Join ^1 cutting AC in 1'. Join RV cutting BG in al 

In the same way we can find as many pairs of points such as^ 
w, a' as we please on BG, the ranges {a) and (a!) will be homo- 
graphic, and their common points will give us vertices of the two 
triangles which can be constructed satisfying the given conditions. 

The process of finding any pair of corresponding points such 
as a, a' is what Chasles calls a construction d'essai^ and if the 
points a, a' happened to coincide, then we should have found one^ 
of the common points*, but if not, the segment aa would be a 
measure of the error, and, as he puts it, we may make use of 
three similar errors given by three trial constructions to solve the 
problem, so that there is a sort of analogy between this general 
method and the arithmetical rules of false position. It will be 
noticed, however, that all that we obtain from such constructions 
in general is three pairs of corresponding points, which constitute 
what we have called the characteristics of the two homographic 
ranges, and even if we chanced to hit upon one pair of correspond- 
ing points which happened to coincide, it would not help us at all 
to find the other pair. ^ 

In Fig. 42 let a, a'; b, b'; c, c' be three pairs of corresponding" 
points obtained as above. Then abc and a'b'c' are the characteristics 
of the ranges, and we might now proceed, as in Art. 82, to rotate 
one of the ranges, construct the cross-axis and find /, J', &c. 
It will be found in practice, however, that this is a long process 
requiring a considerable amount of construction, which increases 
the chance of error in the result, and either / or J' can be 
obtained much more easily and accurately by direct use of the 
fact that it is the position which one of the points a, a assumes 
when the other is at infinity ; in fact, we use the special charac- 
teristics a/oo , a' co' J' instead of the general ones abcj a'b'c. jM 

To find /, suppose a at infinity, and let the line through 1^ 

* We have a similar construction d'essai in Euc. ii, 14; vi, 28; xi, 11. 






92] 



INSCRIPTION OF TRIANGLE 



83 



parallel to BC meet CA in 2'. Draw 2'Q meeting AB in 2. Then 
2P will meet BG in /. 

To find J\ let the line through P parallel to BC meet AB in 
1. Draw 1^ meeting ^C in 1'. Then VR will meet BC in J'. 




Fig. 43. 
In Fig. 43 we have employed Lodge's method, taking the 
points D, D' on opposite sides of IJ\ because la, a' J' are of 
opposite signs. The circle on DD' as diameter intersects BC in 
the common points e, /. Join eP meeting BA in k. Draw kQ 
meeting AC in I. Then le will pass through R, and ekl is one of 
the required triangles. Similarly, starting from the point f, we 
obtain the triangle fnm, which also satisfies the given conditions. 

6—2 



84 CROSS-RATIO GEOMETRY [CH. VIII 

93. In the next two examples we will merely indicate the 
opening steps by which the points /, J' can be obtained, leaving it to 
the student to draw the figures and to find the common points by 
one of the methods given above. Questions of this class might 
also be solved by finding the characteristics, and then projecting 
them on a circle or conic, as will be explained in a later chapter. 
This is the method employed by Cremona in his Mements de 
Geometrie Projective (1875), and in Section xix of that work 
several of these problems are treated in this manner. It should, 
however, be impressed upon a student that, in dealing with any 
given problem, when he has obtained the two homographic 
ranges on a line, and found the three pairs of corresponding 
points, it is not sufficient for him to say "Hence the common 
points of these ranges which satisfy the conditions of the problem 
can be found." For a complete solution the actual positions of 
the common points ought to be determined in every case. 

94. Given two homographic ranges (a) and (a) on two lines 
ALy A'L'j and two points F, P' in the same plane with them, it is 
required to find two corresponding points e, e' in the ranges such 
that the lines Pe and Fe' will contain a given angle <f>. 

Since the ranges are given, if we take any point in the one, we 
are supposed to know the position of its corresponding point in the 
other. Let a, a' be any pair of corresponding points. Join Fa\ 
and through P draw a line making the given angle <f> with P'a' 
and meeting AL in a. Then the ranges (a') and (a') are homo- 
graphic, as are also (a) and (a), and consequently by Art. 39 the 
ranges (a) and (a') are also homographic and we evidently have 
to find their common points. ^ 

To find the point I in the range (a) corresponding to the point ~ 
at infinity in (a). When a' is at infinity. Pa is parallel to AL. 
Through P' draw a line making an angle <f> with AL, and meeting 
A'L' in /'. Then the point / in the range (a) corresponding to /' 
in (a') is the point required. 



J 



93-95] EXAMPLES 85 

To find J\ suppose a is at infinity, and let / be the corre- 
^sponding point in {a). Join j'F^ and through F draw a line 
making with/P' the angle <^. This will meet^Z in the required 
point y, &c. 

95. Given two straight lines Z, L', it is required to find on 
them two points a, a' such that the line aa will subtend given 
angles (f>, cf>' at two fixed 2>oints /*, F, 

On L' take any point A^ and make the angle A-^ PA = ^, and 
A 1 Fa = <li\ the points A and a being on the line L. Then as A^ 
moves along L\ A^ and a trace out homographic ranges, as do ^^ 
and A, and, by Art. 39, A and a. Hence we have to find the 
common points of the ranges (A) and (a'). 

To find I. When a is at infinity, Fa is parallel to L, A^is 
Sit A.^ such that A^Fa = cf>', and A is at /, where A^FI- <f>. 

To find J'. When A is at infinity, FA is parallel to Z, A^ is 
at A^ such that A^FA - <f>, and a is at J', where A^FJ' = <^', &c. 



EXAMPLES. 

1. Determine on a given line a segment which shall subtend given 
angles at two given points. 

2. Determine on a given line a segment of given length which shall sub- 
tend a given angle at a given point. 

3. AIj, AV are two given lines. P is a given point in their plane, and 
a a given point in Ah. Through P it is required to draw a transversal Fhh' 
meeting Ah in 6 and Ah' in V such that Ab' = ab. 

4. In a given triangle inscribe a rectangle equal to a given square. 

5. Given a plane polygon of any number of sides, and the same number 
of points in its plane, inscribe in the polygon another polygon whose sides 
will pass through the given points. 

6. A, B are two fixed points in a given straight line. It is required to 
find in the straight line two other points E, F so that EF may be of given 
length, and the cross-ratio {ABEF) of given magnitude. 



CROSS- RATIO GEOMETRY 



7. Given two homographic ranges on two lines AL, A'L', and 
others on BM, B'M', it is required to draw through a given point P two 
straight lines each of which will intersect AL, A'L' in a pair of corresponding 
points, and will do the same with BM, B'M'. 

[Let a, a' be a pair of corresponding points on AL, A'L', and let Pa meet 
BM in b, and let b' be the point on B'M' corresponding to b. Then if Pb' 
meets A'L' in a', the ranges {a') and (a') are homographic, and the lines 
joining P to their common points are the two lines required. Find I, J', &g.\ 

8. Given a triangle ABC and P a fixed point in its plane, draw through 
P a line cutting AC in m and BC in n such that the triangle mnC may be 
equal to the triangle ABC. 

9. Given a triangle ABC and a point P on the parallel to j5C through A, 
lying between A and the median through B (produced). Draw through P a 
line which will bisect the triangle ABC. 

For other examples of this class the student is referred to 
Chasles, Giom. Siip. pp. 219—223. 
Cremona, Geom. Proj. (1875), pp. 179—188. 
Townsend, Mod. Geom. (1863), vol. ii, pp. 257—275. 



^III ■ 

two ^M 

I 



CHAPTER IX 

INVOLUTION 

96. Def. When two co-axial homographic ranges have the 
Range in points / and J' coincident, the two ranges are said 

Involution. to form a range in involution or an involution 
range. Hence 

A system in involution consists of two co-axial homographic 
ranges^ all that is necessary being that they should he placed so 
that I and J' coincide. 

We must remember that homographic ranges are of two kinds. 
We have (1) those in which / and J' are at a finite distance. 
These are homographic ranges of the second order, and are by 
far the most important, and the involution to which we devote 
our chief attention is when I and J' of such ranges coincide. 
But besides these there are (2) homographic ranges of the first 
order (Arts. 67, 68), when the ranges are divided similarly, and 
both / and J' are at infinity. We shall deal with the condition 
that such ranges shall be in involution in Art. 107. But for the 
present, and in general, we shall confine our attention to ranges 
of the second order, and proceed to discover what special proper- 
ties these possess, when they are in involution. One of the 
most important of these, leading indeed to a second definition of 
involution, follows from consideration of the construction in 
Art. 82, where we may of course select any point we please on 
the double line for the point of rotation, and the two points on 
L and L' corresponding to the point which we select on the 
double line will in general be separate, and the cross-axis which 



88 



CROSS-RATIO GEOMETRY 



[CH. IX 



we obtain by joining them after rotation will have a diflferent 
position for each point selected on the double line, but it will 
always be parallel to the line joining IJ\ Art. 55. 




Fig. 44. 



m' n' P' 



J' 

I 

I 

Fig. 45. 



a'h' c 



L' 



mn p 



Suppose now we have the double line with its two homo- 
graphic ranges, and we slide the line L' along the double line 
until the point J' coincides with /, and then rotate L' about any 
point p (= q) through any angle. 

Let pq' be the cross-axis. Then by Art. 55 pq' is parallel to 
IJ' ; and since p'J' = ql, therefore p'q' = qpj, and if L' is rotated 
back again about p' until it coincides with the double line, the 
point q' will fall upon the point p, and therefore the point ^/ (or q) 
has the same correspondent whether it is considered to be a point 
on L or on L' ; and as p'(-q) may be any point on the double 
line, we may say that 

When two co- axial homographic ranges have the j^oints I and J' 
coincident^ then every point on the axis has its two corresponding 
points coincident. 

This is also clear from consideration of the equation 
Ip ,J'p' = const., 

for if J' coincides with /, and therefore also with their mid-point 
we have Op . Op' - const. = Oe^ = Of^, and the symmetry of the 



96-97] INVOLUTION 89 

relation shews that no matter which range p belongs to, p' is its 
correspondent. The converse is also true; viz. that if any point 
taken on the axis (other than one of the common points) has its 
two corresponding points coincident, the points /, J' will coincide. 
This follows at once from Fig. 44, since if p'q' = pq, we must also 
have p'J' = qlj and therefore when the lines are rotated so that q' 
falls on p, J' must fall on /. Consequently we have a second 
definition of involution, co-extensive with the first, involving the 
same geometrical fact, though emphasising another property, viz. 

Def. Two co-axial homographic ranges are in involution 

when any point on the axis (other than one of the 

Involution common points) has the same corresponding point, 

whether the given point belongs to the first range 

or to the second. This property is given in symbolic form in 

Art. 98. 

The common points must be excluded from this condition 
as each of them corresponds to itself whether the ranges are in 
involution or not. The point taken on the axis will have its 
correspondents coinciding with each other, but not with itself. 
The sufiiciency of the test, and the necessity of excluding the 
common points, are both obvious from Fig. 44. 

97. Def. Since when two co-axial ranges are in involution 
their pairs of corresponding points are interchange- 
Points, able, we will in this case call them conjugate points, 
and the common points of the ranges we will call 
Double double points. This distinction will serve to shew 
the reader whether we are speaking of two co-axial 
homographic ranges in general, or in the special case where they 
form a system in involution. 

Since /, J' coincide, they also coincide with their mid-point 0. 

Def. The point is now called the centime of 
the system*. 

* Chasles, Apergu Historique, p. 318. 



90 CROSS-RATIO GEOMETRY [CH. IX 

When two ranges form a system in involution we will denote] 
it by {aa', bb\ cc ...), where aa', bb'j cc ... are pairs of conjugate 
points, and we shall in general use the letters e, f in speaking of] 
the double points. 

98. When we have a system in involution, if we take three^ 
pairs of conjugate points and consider four of the six points 
(provided they do not form two pairs of conjugates), it is easy to 
see that their cross- ratio is equal to that of their conjugates*. | 

Thus if a, a ; 6, 6' ; c, c are the three pairs of points, we may 
take a, 6, c, a as belonging to the first range, and then by the 
secondary definition of involution given in Art. 96 it follows that 
a', 6', c', a are their correspondents in the second range, so that 
{abca') = (a'b'c'a), or referring to Fig. 45, 

(abca') = (abcm) since m and a coincide 

= iab'c'm) since the rows are homographic 

= {a'b'c'a) since m! and a coincide. 

This property is the symbolic form of the definition of Art. 96, 
ad fin. which established two converse theorems, viz. that if 
a, a'-, b, b'; c, c' are in involution, then (abca') = (a'b'c'a), and 
similarly for other sets (see equations 1 — 7, in Art. 106), and'^ 
conversely, if 

(abca') = {a'b'c'a), 

then a, a ; b, b' ; c, c are pairs of points in involution. It will^ 
be found that this property is in many problems the easiest 
involution property to discover. See also Art. 105. ^ 

99. We now come to a third property of involution, deducible ^ 
at once from Art. 98. Since {aa'ef) = {alaef), the range aa!ef 
by Art. 28 is harmonic, and the double points are harmonic i 
conjugates for each pair of conjugate points. 

* Chasles, Apercju Historique, p. 313. 



J 

>> 
[f 
d 

1 

"I 



98-101] ONE DOUBLE POINT AT INFINITY 91 

As a special case if one of the double points is at infinity, the 
other bisects each of the segments joining the pairs of conjugates. 
When this case occurs, the homography is of the first order, and 
is discussed in Art. 107. 

100. The property given in Arts. 96, 98 may also be seen 
in Fig. 46 where the ranges form a system in involution, 
centre 0. Small letters denote points in the upper range, and 
their conjugates are denoted by accented capitals, so that (a, A') 
represent a pair of conjugates, as do also (a^, A^'). 

a b bj Ui d c Cj d^ 

Ai B\ B' A' b b'l C'lC D' 

Fig. 46. 

The relations Oe^ = Of^ = Oa . OA' = Oa, .0A,'=... shew that as 
regards any pair of conjugates it is immaterial which of them we 
assign to the upper, and which to the lower range, so that we can 
either say Oa . OA' or OA^' . Oa^ ; i.e. any point has the same 
conjugate whether it belongs to the upper or the lower range, and 
any pair of conjugate points (a, A') give rise to another pair of 
conjugates (a^, A^'), which coincide with the former pair when 
taken inversely. 

The same result can be deduced from the following proposi- 
tion. 

101. In two equicross ranges, if {a, A') are any pair of 
corresponding points, we can always find another pair of corre- 
sponding points («!, J/) such that the segments aa^ and A' A( are 
of equal length. 

a I a\ 

~A\ T A* 

Fig. 47. 

For on the upper range take a point a-^ such that la^ — J' A'. 
Then the correspondent of a-y is yl/, where 



[CH. IX 



92 CROSS-RATIO GEOMETRY 

la^ . J'A^' = la . J' A', 
.-. J'A; = Ia, 
.'. al + la^ — A^J' + J'A\ i.e. aa^ = A^'A'. 
Also aAj' = a^A' = IJ' . 

Hence if we are given any two co-axial homographic rows, and 
if we move one of them (say the accented row) along the other 
until the points /, J' coincide at 0, a will coincide with A(^ 
and «! with A\ and as a is any point on the axis, we have the 
property of the previous Article. 

102. Given two pairs of conjugate points a, a'; b, b' ; to find 
the centre of the involution. 




Fig. 48 




Fig. 50. 



102-104] CONSTKUCTION OF CENTRE 93 



I 



From the relation Oa . Oa' = Oh . Oh' we have 
Oa Oh' Oa + ah' ah' 



Oh Oa' Oh + ha' ha' 
Through a and h draw any pair of parallel lines aa, hjB in the 
same or opposite senses, according as ah', ha are in the same or 
opposite directions, and take aa = ah', and hjS = ha'. Then the 
line joining aft will meet the axis in the required point 0. 

103. The value of the expression Oa . Oa is called the power 
of the involution, and is easily found in terms of the segments 
between the points a, a', h, h'. 

For, as in Art. 102, 

Oa^^,.Oh' = ~ (Oa + ah'), 
ha ha ^ ' 

.'. Oa (h'a' - ah) = ah . ah'. 

Again, Oa' = —7- . Oh = —j (Oa' + ah), 

.'. Oa'{ah-h'a') = h'a'.a'h, 
ah . ah' . ah . ah' 



Oa. Oa' 



{ah + a'h'Y 



104. Just as three pairs of points are sufficient to determine 
the homography of two ranges, two pairs of points are sufficient 
to determine an involution, and may be called its characteristic. 
For if a, a'; h, h' are two given pairs of points, we can by 
Art. 102 find the point 0, and the involution condition will then 
determine the conjugate c' of any point c, which we can find by 
Euc. VI, 12, from the relation 0c.0c' = 0a. Oa'. Therefore if 
six points are in involution there is a relation between them from 
which, if five are given, the sixth can be found ; and three pairs 
is the least number between which this relation can exist. It 
must be remembered that since two pairs determine the involu- 
tion, if a, a' ', h,h' ; c, c are in involution, and also a, a' ; h,h'; d, d', 
it follows that a, a' ; h, h' ; c, c ; d, d' form a system in involution. 



94 CROSS-RATIO GEOMETRY [CH. IX 

105. If we have given six collinear points which are con- 
nected by an equation of cross-ratios, the following consideration 
enables us to determine by inspection of the equation whether 
the points are in involution, or not. 

If from six points ^;, q, r, s, t, u we can form two equicross 
ranges, such that of the two points (say q, u) which are necessarily 
common to both ranges each has the other for its correspondent 
wherever they occur, the six points are in involution. Thus, if 
(pqru) = (tusq), then p, t ; q, u and r, s are three pairs of points 
in involution. 

Of course we may have to rearrange one of the cross-ratios 
before it takes the requisite form, but if we can by interchanging 
pairs of letters in accordance with the rule of Art. . 3 put one of 
the members into the requisite form whilst retaining the equality 
of the cross-ratio, there is involution, e.g. Suppose we are given 
{pqru) = (utqs). The right-hand side can be written in the form 
{tusq), and therefore {pqru) =■ {tusq), shewing that there is in- 
volution since the repeated letters now correspond. If we cannot 
do this, there is not involution, as in {pqru) — {qust) or = {stqu). 

The necessary and sufficient condition for involution is the 
following rule: 

Whatever places the re2)eated letters occwpy in one of the cross- 
ratios, both or neither of them must occujyy these places in the 
second cross-ratio. 

Important Involution Equations. 

106. Since aa'ef is a harmonic range, 

.-. {aaef)=- 1, 

. ae a'e 

" af^~<f' 

and referring this to any arbitrary origin m, 

ine — 7na me — mcs' 



jnf— ma i^f— 'nia ' 
.'. {ma + ma'} {me + 7nf) = 2ma . ma + 2me . mf 



105-106] INVOLUTION EQUATIONS 95 

and if 0, a are the mid-points of e/J ««', 

2ma . mO = ma . ma' + me . mf. 
Similarly if /8 is the mid-point of hh' y 

2my8 .mO = mh . mh' + me . mf, 
.'. 2a/3 . 7nO = mh . mh' — ma . ma *, 

with two similar equations obtained by introducing y the mid- 
point of cc'. 

If m coincides first with a, and then with a', we obtain 
(1) 2ap.aO = ab.ab'f, (2) 2al3 .a'O = ab . ab'. 

Similarly we have 
(1) 2ay .aO = ac.ac, (2) 2ay . a'O -a'c .a'c, 

ap ab . ab' a!b . a'b' 



i.e. (aa'bc) = (aab'c) ...(1). 

ay """"""" " ^ / \ ' 



Similarly 

I3y 



/a 



ac . 


ac 


a'c . 


a'c" 


be 


.be' 


b'e 


. b'e' 


ba. 


.ba 


~ b'a 


.b'a" 


ca . 


ea 


ca. 


c'a' 


cb\ 


eh' 


~ c'b. 


e'b" 



ISa~^^ ;../-/.'. A'." i.e. {bb'ca) = {b'bc'a) ...{2), 



and -^ - '^'■^'l^ _ " ^' ' '^ 7 . {,e. lee'ab) - (e'ca'b') ...(3). 

yf3 cb .cb cb.eb ^ ' ^ ' ^ 

By suitable multiplication we may obtain the properties 

ab' . be' .ea' = — a'b . b'e . e'a | or {abe'a') = (a'b'ca) . . . (4), 

ab' . be . e'a' = - a'b. b'e' . ea or (abca) = (a'b'c'a) . . . (5), 

ab .b'e .ca' — — a'b' .be .c'a or {ab'c'a) = {a'bca) ...(6), 

ab . b'e . c'a = — a'b' . be . ca or {ab'ea) = (abca) • . • (7). 

Any one of these seven equations expresses the condition that 
must hold when the six points a, a ; b, b' ; c, c are in involution, 
and consequently from it each of the other six equations can be 
obtained. These results of course follow directly from the 
definitions of Art. 96. The object of giving them here is to 
call attention to the fact that the principles involved in the 

* Pappus, Bk VII, Props. 45 — 56. f Pappus, Bk vii, Prop. 41. 

X Pappus, Bk VII, Prop. 130. See also Appendix I. 



96 CROSS-RATIO GEOMETRY [CH. IX 

definitions of involution can be just as readily obtained from 
properties given by Pappus. 

107. Centre of the involution at infinity. 

If we consider the distance between the double points e, f to 
gradually increase, so that, while e remains at a finite distance 
f recedes to infinity, the centre of the involution, which by 
Art. 72 is midway between them, is also at infinity. As this 
point is formed by the coincidence of / and J', it follows that 
the ranges are similar, but we shall see that the involution 
condition of Art, 99 makes them not only similar, but identical, 
in opposite senses. For when f is at infinity, e bisects each of 
the segments aa\ bh', .... Hence ea = — ea', eb = — eb\ . . . , 
.'. ah = — ah', and similarly hc = — h'c'y &c. 

Consequently, when the centre of the involution is at infinity, 
and one of the double points is at a finite distance, while the 
other is at infinity, the two ranges are identical, but in opposite 
senses, and the finite double point bisects each of the segments 
joining pairs of conjugate points. 

This is the only practical case of involution of the first order, 
for if both double points were at infinity, every point on the 
range would have its conjugate at infinity. Consequently, similar 
ranges cannot be in involution unless they are identical, and in 
opposite senses, and then they are always in involution, for on 
drawing a figure it will be seen that the point which bisects the 
segment joining one pair of conjugate points necessarily bisects 
every other pair. 

To avoid repetition in future we would remark that in all 
cases of involution the ranges are supposed, unless it is otherwise 
stated, to have their points /, J' coincident at a finite distance, 
and consequently their homographic relation is of the second order. 

108. Since Oa. Oa' = 0b . Oh' = ... = 06^=0/", 4 

it is evident that if any pair of conjugate points such as a, a' are 
on opposite sides of 0, the product Oa . Oa! is negative, am' 



J 



107-108] 



DOUBLE POINTS 



97 



therefore the double points e, / are imaginary. When this is the 
case for any pair of conjugates, of course it must hold for every 
pair, for if the product of any pair Oa . 0(i is negative, that of 
every pair must also be negative. 




Fig. 51. 

A little consideration of Fig. 51 will shew that when the 
point divides the segments act', W internally, if Oa > Oh, then 
Oa < Ob', and therefore the segments aa', bb' overlap, and this will 
obviously be the case with all the segments joining pairs of 
conjugates, and the double points are imaginary. 




Fig. 52. 



98 CROSS-RATIO GEOMETRY [CH. IX 

On the other hand the double points will be real when the 
product of any pair such as Oa . Oa! is positive. The different 
positions where this can be the case are shewn in Fig. 52, in 
which, if we consider any pair of the segments act ^ hh', gc\ dd' we 
see that any segment lies entirely within or entirely without the 
other, and that no two segments overlap. Hence we have the 
simple rule : 

The double points are real or imaginary according as the 
segments joining pairs of conjugate points do not or do overlap. 

This might also be shewn by using Lodge's method, Art. 84. 
Thus in Figs. 51, 52, if we draw Oa, Oa perpendicular to the axis 
equal to Oa, Oa' respectively, and on opposite sides of the base if 
the product Oa . Oa is positive, and on the same side if it is 
negative, the double points will be the intersection of the line 
and circle. 

Lodge's circles also give us a simple method of constructing 
the conjugate of any point m on the base. For we have merely 
to join ma cutting the circle in M. Then aM will meet the base 
in the required point m'. The construction holds whether the 
double points are real or imaginary. 

109. If we rotate the accented row about the point through 
two right angles it will still be homographic to the other, and will 
form with it another system in involution ; and since each of its 
points of division will now be on the opposite side of to what it 
was before, we see that if the first system is an overlapping one, 
the second is a non-overlapping one, and vice versa. 

One-to-One Correspondence. 

110. If we take the general relation given in Art. 70 and 
express the condition that I and J' should both coincide with 0, 
we have 

pm . pm' —pO (pm + ptn) +pO . pp' = 0, 



I 



109-111] INVOLUTION PENCILS 99 

which is of the form 

a^' + h {x + x') + l = 0. 

This gives us a one-to-one correspondence in which x and x' are 
interchangeable, and is merely another way of expressing the fact 
that the two series of points given by the equation have the 
property mentioned in Art. 96 and are in involution. 

If the homographic equation is of the first order, if x and x' 
are to be interchangeable, it must be of the form 

x + x' = K, 

in which case one of the double points is at infinity, and the other 

at a distance ^ from the common origin. 

Involution Pencils. 

111. Def. If the divisions of an involution range are joined 
Involution to an external point, the pencil so formed is called 

Pencil. an involution pencil. 

When two concentric pencils form a system in involution, we 
will denote it by V{aa, hb\ ...), where Fa, Va' are a pair of 
conjugate rays, and Ve, Vf will in general be used to indicate 
the double rays. 

By Art. 21 any transversal will cut an involution pencil 
in an involution range, and the double rays of the pencil will 
cut the transversal in the double points of the range. 

By Art. 31 the angle between any pair of conjugate rays is 
divided harmonically by the double rays; and conversely, if aVa\ 
b Vb', c Vc are three angles which are all divided harmonically by 
the same pair of lines Ve, Vf, then V {aa', bb', cc') is an involution 
pencil having Ve and V/ior double rays. 

It should be noticed that there is no ray which can be called 
the central ray of the pencil, and in that respect it differs from 
an involution range. The ray conjugate to that drawn parallel 

7—2 



100 CROSS-RATIO GEOMETRY [CH. IX 

to the range passes through the centre of the range, and of course 
this ray will be different for different transversals, except when 
the transversals are parallel. 

112. In an involution pencil there exists one, and in general 
only one, pair of conjugate rays at right angles. When there is 
more tJian one, every pair of conjugate rays intersect at right 
angles. 

Let the pencil, vertex P, cut any transversal in the pairs of 
conjugate points a, a' ; b, b' ; ..., and let be the centre of the 
involution range on the transversal. Join FO, and on PO, or PO 
produced, take a point Q such that PO . OQ^aO . Oa — 

Then every circle through the two points P, Q will cut the 
transversal in a pair of conjugate points. There will be in 
general one and only one such circle having its centre in the 
transversal ; and this alone will cut it in two conjugate points 
c, c which will subtend a right angle at P. If, however, PO —- OQ, 
and PQ cuts the transversal at right angles, every such circle 
will have its centre in the transversal, and all pairs of conjugate 
rays will be at right angles. 

It follows from the above that 

If any number of right angles have the same vertex, their sides 
fortn an involution pencil. 

Such a pencil may be called orthogonal. 



Circular Points at Infinity. 

113. Since the involution range formed by an orthogonal 
pencil on any transversal is overlapping, the double rays of an 
orthogonal pencil are imaginary. If the rays of the pencil are 
produced to meet the line at infinity, they determine on it an 
involution range of ideal points with imaginary double points. 
These double points, though imaginary, are of very great import- 

C£PART,V,E T Or- MATHEMATICS 
.. liNIVERiiry OE TORONTO 



I 



112-114] CIRCULAR POINTS ' 101 

ance in connection with the subject. They are in a sense unique. 
For any two orthogonal systems each containing an infinite 
number of rays are superposable by mere translation without 
rotation, and parallel rays of the two systems will correspond to 
each other. These parallel rays intersect in the line at infinity, 
as do also their corresponding double rays, so that we may 
consider that all orthogonal systems determine the same involu- 
tion range of ideal points on the line at infinity, and consequently 
the double rays of every orthogonal system pass through the same 
pair of imaginary points on this line. 

For reasons that will be given in Chap. XIII these points are 
called the ch'cular points at infinity, or shortly, the circular points, 
as it will be shewn there that all circles pass through them. The 
lines joining a real origin to these points are called isotropic lines, 
and their equations are 2/ = + ix, where i is the imaginary quantity 
V- 1, and as these points are imaginary and lie on these lines we 
will denote them by the letters i, i'. At present we wish merely 
to direct the student's attention to the fact that they may be 
considered as perfectly definite, though imaginary; their connec- 
tion with circles and conies will be discussed later in Chap. XIII 
and more fully in Chap. XIX. Here we are not attempting 
to give any rigid proofs of their uniqueness or their properties, 
but we thought it would be interesting to the student to have 
his attention drawn to this pair of remarkable points. 

114. If V is the vertex of an orthogonal pencil, and Va, Va' 
a pair of conjugate rays, the pencil V {aa'ii') is harmonic, and 
conversely, if we are given that the pencil V {aa'ii') is harmonic, 
tlie angle aVa is a right angle, (By Art. 113.) 

The latter part of this property may be stated: An involution 
pencil having the isotropic lines for double rays is orthogonal. 



102 CROSS-RATIO GEOMETRY [CH. IX 



EXAMPLES. 

1. If two ranges are homographic, and any point P on their cross-axis 
is joined to the points of division on the ranges, these rays will form two 
pencils in involution. 

[For if the ranges intersect in the point A, then in both pencils the cross- 
axis will have PA for its corresponding ray.] 

2. If e, f are the double points of the involution whose characteristic is 
rt, a' ; b, b\ shew that {ab', a'b, ef) form a system in involution, also 
(a&, a'b', ef) form a third system in involution, whose double points, if real, 
are conjugate points of the first system. 

Of these three systems formed by taking all possible pairs of the character- 
istic points a, a', 6, &', two have real double points, and one has them 
imaginary, and the double points of each system are conjugate points of the 
other two. 

3. Given two homographic pencils, centres 0, 0', shew that any trans- 
versal through the cross-centre T will be cut by the pencils in two ranges in 
involution. 

In Fig. 27, p. 43, let any transversal through T cut 00' in P. Then in 
both ranges P has T for its corresponding point. 

4. Three fixed points A, B, G are given on a straight line, on which two 
other points D, E are taken so that 

{ABCD) = \ and MBCE)=^^,, 

where a, h, «', h' are constants, and X a variable parameter. 

Shew that D, E will be conjugate points of an involution if a^b' = Q. 
[Let AG=p, BC = q, .'. AB=p-q, 

AD = x, .: BD = x-p + q; AE = y, .'. BE = y-p + q, 



a'\ + b' ' ' AE BE y y-p + q 

Substituting for X we obtain a relation between x and y of the form 
Pxy + Q(x + y) + R = 0.] 



CHAPTER X 

INVOLUTION AND HARMONIC SECTION. HARMONIC PROPER- 
TIES OF A QUADRANGLE AND QUADRILATERAL. POLE 
AND POLAR 

Relation between involution and harmonic section. 

115. One of the most important properties connected with 
a system in involution is that of Art. 99, which tells us that 
{aa'ef) is a harmonic range, so that if aa\ bb', cc' are pairs of 
conjugate points forming a system in involution, of which e, f 
are the double points, we may say that the axis of the involution 
is harmonically divided at the points cm'^ bb\ gc\ . . . for the points 
e,/, and it follows by Art. 108 that when two segments are harmonic 
conjugates for a third segment^ one of them is entirely within or 
entirely without the other when the third segrnent is real; but if the 
third is im^bginary, the other two will overlap. 

AlsOf given three pairs of points aa', bb\ cc in involution^ if 
e, f are harmonic conjugates for a, a' and b, b', then e, f are the 
double points of the involution, and are therefore harmxynic con- 
jugates for c, c . 

116. By Art. 110 we see that if axu! is any given pair, and 
miri a variable pair of conjugate points, 

am> . a»i' — aO {am, + am') + aO . aa' = 0, 



1 



104 CROSS-RATIO GEOMETRY [CH. X 

and when m and m' coincide, the double points are given by 

am? — 2aO . am + aO . aa' = ( I ). 

Now whether the system is overlapping or non-overlapping, 
the point is always real, for its position can always be found by 
Art. 102. Therefore the product aO . aa is also always real, and 
consequently the product of the roots of the equation (1), i.e. the 
product ae . af, is always real. 

Again, suppose we have given two real points e, f on a line, if 
we take any other point a on the line we can always find a its 
harmonic conjugate for e andy, i.e. we can find an infinite number 
of pairs of conjugate points aa\ bb\ cc\ ... such that each pair 
taken with ef forms a harmonic range; and we are merely ex- 
pressing the same geometrical fact in a different way when we say 
that aa, hh\ ... form a system in involution in which e,f are the 
double points. Also, in the case where e,/ are a pair of imagin- 
ary points on the line, and a any real point on it, if we know 
the position of 0, and the value of the product ae . af^ i.e. the 
value of aO . aa\ we can always find a the harmonic conjugate 
of a for e, f. So that we can either commence with a system 
in involution, and proceed to find the double points, which will 
be imaginary or real according as the system is overlapping, or 
non-overlapping; or we can begin with the double points and 
from them construct the involution system which will be over- 
lapping or non-overlapping according as the double points are 
imaginary or real. 

In Fig. e02, if we suppose the range carrying the accented 
letters to rotate about through two right angles, so as to bring 
a', h\... into the positions a", />", ..., we shall have 



Oa . Oa!' = Ob . Ob" = ... =-0e' = - 0/^ 
^OE"=^OF\ 

The points a, a" ; b, b"; ... will be pairs of harmonic conjugates 
for the imaginary points U, F, whose mid-point is real. 



116-117] QUADRILATERAL AND QUADRANGLE 105 

L 





Fig. 54. 

117. As we shall frequently in the following pages have to 
Quadrilateral, deal with quadrilaterals and quadrangles, we will 
Quadrangle. remind the student that a quadrilateral is a col- 
lection of four lines, no three of which pass through the same 
point. If we call the intersection of any two of these lines a 
vertex, there are three pairs of opposite vertices, the lines joining 
which are called diagonals, or diagonal lines, and form the sides 
of a diagonal triangle, and the whole figure is called a complete 
quadrilateral. 



106 CROSS-RATIO GEOMETRY [CH. X 

A quadrangle is a collection of four points, no three of which 
lie on the same straight line. If we call the line joining any two 
of these points a side, there are three pairs of opposite sides, the 
intersections of which are called diagonal points and are the 
vertices of a diagonal triangle, and the whole figure is called 
a complete quadrangle. 

Fig. 53 shews us the complete quadrilateral with its 4 lines, 
6 vertices, and 3 diagonal lines joining pairs of opposite vertices 
forming the diagonal triangle GG'H. 

In Fig. 54 we have the complete quadrangle with its 4 points, 
6 lines and 3 diagonal points forming the diagonal triangle GEF. 

In Fig. 55, if ABCD is taken as a quadrangle, its diagonal 
triangle is GEF, but if we consider ABCD as a quadrilateral its 
diagonal triangle is GG't. In the same figure it will be noticed 
that in the quadrilateral and the quadrangle the diagonal triangles 
have the same vertex G, and their bases are in the same straight 
line. It is arbitrary which quadrilateral (of three) we take 
in conjunction with a given quadrangle, and this accounts for 
the want of symmetry in the relations between the two. 



Harmonic properties of a quadrangle 
and quadrilateral. 

118. In a complete figure such as Fig. 55: 

(1) In the quadrangle ABCD the three pairs of opposite sides 
cut liarmonically the three sides of its diagonal triangle GEF, 
i.e. the ranges (EsGr), (FqGp), (FG'Et) are harmonic, 

(2) In the quadrilateral ABCD the three pairs of opposite 
vertices divide hannionically the three sides of its diagonal triangle 
GG't, i.e. the ranges (CGAt), (DGBG'), (FG'Et) are harmonic. 

The extremities of the bases EF, tG' of the two diagonal 
triangles form a harmonic range (FG'Et). 



118] QUADRILATERAL AND QUADRANGLE 107 

[It will be shewn also that ever}' line in the figure is divided 
harmonically. The proofs of the foregoing properties are given 
without reference to any special order. Owing to the particular 
quadrilateral that has been taken in conjunction with the quad- 
rangle the third range in (1) is identical with the third range in 
(2), while the others are distinct.] 




The ranges (tAGC), {tEG'F) are in perspective, centre D; 
therefore by Art. 21 their cross-ratios are equal. Also the ranges 
(iAGC)f (tFG'E) are in perspective, centre B. Therefore their 
cross-ratios are equal. 

.-. {tEG'F) = {tFG'E), 

i.e. the range (tEG'F) being unchanged in value when the pair of 
points E, F are interchanged separately, is harmonic by Art. 28, 
and, consequently, so also is {tAGG\ and F(tAGC) is a harmonic 
pencil, which therefore cuts every transversal in a harmonic range 
by Art. 21, and hence the following ranges are all harmonic, 
viz. (DpAE), (rGsE), (CqBE) and (DGBG). 



108 



CROSS-RATIO GEOMETRY 



[CH. X 



Similarly, since E {DGBG') is a harmonic pencil, the following 
are all harmonic ranges, viz. (AsBF), {pGqF) and (DrCF). 
We have proved incidentally the property 

When two equicross ranges of four points intersect in one of 
the pairs of corresponding points, the condition that tJiey should be 
harmonic is that they should he in perspective at two different 
centres. 

Of course, if two ranges are in perspective, and if we know 
that one of them is harmonic, the other will also be harmonic 
by Art. 21. 




Fig. 56. 



The results of this article also give us a method of drawing 
a fourth ray of a harmonic pencil when three of them are given. 

(1) Suppose we have given the pairs OC, OD and OA. 
Through A draw two transversals ACD, Acd. 

Join Cd, cD, intersecting in B. Then OB is the conjugate 
of OA. 

(2) Suppose we have given OC, OD and OB. 

On OB take any point B, and through it draw two trans- 
versals cBD, CBd. Join cd, CD, intersecting in A. Then OA 
is the conjugate of OB. 



118-120] QUADRILATERAL AND QUADRANGLE 



109 



119. Any transversal drawn across a quadrangle cuts its 
three pairs of opposite sides in six j^oints which form a system in 
involution*. 




Fig. 57. 

Let L be the transversal, and let it cut the equal pencils 
A {BGc'D) and G {BGc'D) in the ranges acc'h' and bcc'a'. 

Then {acc'h') = (bcc'a') = (a'c'cb) by Art. 3. 

Therefore by rule of Art. 105, aa'y hb\ cc are six points in 
involution. 

120. The six lines drawn from any point to the three pairs of 
opposite vertices of a quadrilateral form a pencil in involution. 

In Fig. 57 let be the ^iven point, and take the pencils 
joining it to the ranges {AEe'D\ (BFeC), which are in per- 
spective. 

We have (AEFD) = (BEFG) = {GFEB). 

Therefore by the rule of Art. 105, the pencil {AG, BD, EF) 
is in involution. 

* Pappus, Bk VII, Prop. 130. See also Appendix I, p. 124. 



110 CROSS-RATIO GEOMETRY [CH. X 



Pole and Polar. 

121. Another relation which is intimately connected with 
harmonic section is that of pole and polar, which we shall find of 
the greatest importance when we come to treat of conies. For the 
present we give the following : 

Def. If we have an angle BAC and any point in its plane, 
and if we draw AP the harmonic conjugate oi AO 
for the lines AB, AC, then AP is called the polar 

of 0, and is called the pole of AP for the lines AB^ AC or 

for the angle BAC. See Fig. 58. 

122. The polars of a point for two angles of a triangle 
intersect on the line which joins the point to the thii'd vertex of 
the triangle. 

A 




Let be the given point, AP the polar of for the angle A, 
and BP the polar of for B. Then shall CO pass through P. 

For the pencils A {COBP) and B {CO A P) are harmonic, and 
consequently equicross, and they have the common ray AB. 
Therefore, by Art. 25 their intersections C, 0, P are collinear. 

123. Let AO, BO, CO be any three concurrent lines through 
the vertices of a triangle, meeting the opposite sides in a, y8, y 
respectively, and let (iy meet BC in a. Then Aa is the polar of 
for the angle BAC. 



121-124] 



POLE AND POLAR 



111 



For the ranges (BaCa) and (ySySa') are equicross, being in 
perspective, centre A. And the ranges {BaCa') and (ySSya') are 
equicross, being in perspective, centre 0. 

Therefore (y3/?a') = (^SSya'). 




Fig. 59. 



Therefore by Art. 28 the range (yS^a) is harmonic, and there- 
fore so also is the range (BaCa) i.e. Aa is the polar of for AB, AC. 

Similarly, if ya, a^ meet CA, AB respectively in y8', y, B(i' is 
the polar of for the angle ABC^ and Cy is the polar of for 
the angle ACB. 

124. The polar s of a given point for the three angles of a 
triangle meet the opposite sides in three points which are collinear. 

Let be the given point in Fig. 59, and produce AO, BO, CO 
to meet the opposite sides in a, ^8, y, and let the sides of the 
triangle a^y be produced to meet BC^ CA, AB in a, p\ y 
respectively. Then Aa\ Bp\ Cy are the polars of for the 
angles Ay B, C. 



112 CROSS-RATIO GEOMETRY [CH. X 

Now the triangles a^y, ABC being co-polar, are co-axial, by 
Art. 26. Therefore a\ yS', y are collinear. 

Conversely, if a transversal meets the sides BC, CA, AB of a 
triangle in a', yS', y', and on these sides we take a, ^, y the harmonic 
conjugates of a, fi', y respectively, the lines Aa, B/3, Cy are con- 
current. 

CoR. The sides BC, CA, AB are harmonically divided at 
a, a ; /3, p' ', y, y ; and these six points lie by threes on the four 
straight lines a'yS'y', a'^y, a^ y, aySy'. 

125. In Fig. 67 if Q is any point in the plane of the quad- 
rangle ABCD, the polar s of Q for the angles DEC, AFD, DGC are 
concurrent. 

Let EQ' be the polar of Q for the angle AEC, and FQ' its 
polar for AFC. Join QQ' cutting the three pairs of opposite 
sides of the quadrangle in the points aa, hh', cc, which by Art. 119 
form a system in involution. 

Then since by Art. 31 Q, Q' are harmonic conjugates for a, a, 
and also for h, h', they are the double points of the system in 
involution to which a, a' and h, h' belong. But c, c' belong to the 
same system. Therefore by Art. 115 Q, Q' are harmonic con- 
jugates for c, G, i.e. the polar of Q for the angle CGD passes 
through the point Q'. 

EXAMPLES. 

1. The lines joining the vertices of a triangle to the mid-points of the 
opposite sides are concurrent. 

[In the converse of Art. 124 suppose the transversal to be at infinity.] 

2. In Fig. 57 let the transversal L meet EF in /, and on EF take X the 
harmonic conjugate of I for the points E, F. 

Similarly on AC take 7 the harmonic conjugate of c for ^, C, and on BD 
take 7' the harmonic conjugate of c' for j5, D. Then will X, 7, 7' be collinear. 

[Let X7 meet BD in 7", and L in P. Then by Art. 120, P (EF, AC, BD) 
is an involution pencil; and by construction P (EF, l\) and P (AC, cy), i.e. 
P(AC, l\), are harmonic. .*. P (BD, l\), i.e. P(BD, c'y"), is harmonic. 
Therefore 7" coincides with 7'.] 



I 



125] EXAMPLES 113 

3. The mid-points of the three diagonals of a complete quadrilateral are 
collinear. 

[In Fig. 57 let A BCD be the quadrilateral, AG, BD, EF its three 
diagonals. In Ex. 2 let the transversal L be the line at infinity. Then 
Z, c, c' are at infinity, and X, 7, 7', the mid-points of EF, AG, BD, are 
collinear.] 

4. Shew that the three points in which the external bisectors of the 
angles of a triangle meet the sides produced are collinear. 

5. The corresponding sides be, b'c', etc. of two triangles abc, a'b'c' in 
plane perspective intersect in a, /3, 7 respectively, and aa', bb', cc' respectively 
intersect the line a/37 in a', /3', 7'. 

Prove that the range (aa', /3/3', 77') forms a system in involution. Use 
Fig. 12, p. 20. 

6. ABG is a triangle, and O any point in its plane. Prove that the 
external bisectors of the angles BOA, AOG, GOB intersect the sides AB, AG, 
BG respectively in three points which lie on a straight line. 

7. The lines OA', OB', OG' bisect the internal angles formed by the lines 
joining any point to the angular points of the triangle ABG, meeting BG 
in A', GA in B' and AB in G'. Also A", B", G" are harmonic conjugates of 
A', B', G' for B and G, G and A, A and B. Prove that A", B", G" are 
collinear. 

8. If a circle is described about a triangle, the points where the tangents 
at its vertices meet the opposite sides are collinear. 

9. Prove that if ABG, BEF are two triangles, and if <S is a point such 
that SD, SE, SF cut the sides BG, GA, AB respectively in three collinear 
points, then SA, SB, SG will cut the sides EF, FD, DE respectively in three 
points which are collinear. 

10. If through 0, the intersection of the diagonals of a quadrilateral 
ABCD, a line OH is drawn parallel to the side AB meeting GD in G and the 
third diagonal in H, prove that OH is bisected at G. 



M. 



APPENDIX I 

pappus' account of the PORISMS of EUCLID, AND HIS 
LEMMAS (l — XIX) ON THEM 

I PROPOSE to give in this Appendix a short account of Euclid's 
Treatise on Porisms, the loss of which is probably more to be 
regretted than that of any of the other treatises on geometrical 
analysis which were extant in the time of Pappus (400 a.d.) 
but which have since disappeared. The reason why reference 
should be made to it in the present work will be obvious to anyone 
who glances in the most cursory manner over the Lemmas which 
Pappus gave as explanatory propositions to it. From them one 
can hardly fail to draw the conclusion that the master mind 
which conceived the Porisms was quite familiar with the funda- 
mental principles of homography, I mean with harmonic section, 
the harmonic properties of a quadrilateral, homographic ranges 
and pencils, and involution. Unfortunately the ancient geometers 
suffered from three hindrances, viz. ( 1 ) the non-recognition of con- 
tinuity, (2) the non-recognition of sense in the direction of lines 
and description of angles, and (3) the absence of a suitable 
notation. It is due in a great measure to the removal of these 
fetters that homography has been enabled to make the strides 
which it has done in the last 100 years. 

As our knowledge of the Treatise on Porisms is almost con- 
fined to the description given by Pappus in the preface to the 
Seventh Book of the Mathematical Collections, and as this has 
not up to the present time, as far as I am aware, been given in 



pappus' account of Euclid's porisms 115 

English, I have thought it advisable to give a translation of 
Pappus' account in the hope that it may not only serve the pur- 
pose of allowing the student to obtain his information on the 
subject from the fountain head, but may also perchance be the 
means of enabling some one versed in Oriental languages to 
recognise a copy, perhaps in an Eastern dress, in one of our 
public or private libraries, where it is quite possible that one may 
be lying involutus pulvere magis quam tenehris sids*. 

\Translation.'\ 
" The Three Books of Porisms. 

"After the books on contacts (by Apollonius) come the Porisms 
of Euclid in three books, a most ingenious collection for solving 
more difficult problems of which the nature of the subject 
provides an unlimited number. No addition has been made to 
them as Euclid first wrote them, except that certain stupid 
persons before our time have given alternative versions in the 
case of a few of them, for each porism has a definite number of 
ways in which it can be stated, as we have pointed out, and 
Euclid has given only one in each case, and that the most 
obvious one. They are in principle subtle and natural, and in- 
dispensable and quite general, and afford much pleasure to those 
who are able to understand and investigate them. 

Porisms of all classes are neither theorems nor problems, but 
they occupy a position intermediate between the two, so that 
their enunciations can be stated either as theorems or problems, 
and consequently some geometers think that they are really 
theorems, and others that they are problems, being guided solely 
by the form of the enunciation. But it is clear from the defi- 
nitions that the old geometers understood better the difference 
between the three classes. For they said that a theorem is that 

* Cf. p. 75. 

8—2 



116 CROSS-RATIO GEOMETRY [APP. I 

in which something is proposed for demonstration, a problem is 
that in which something is proposed for construction, and a 
porism is that in which something is proposed for [discussion or] 
investigation. This definition has been changed by later writers, 
who were not able to fully investigate them, but as is usual in 
the Elements [of Euclid] they only gave a demonstration of the 
quaesitum, without also giving a discussion of it. And although 
they are shewn to be mistaken by the definition given above, and 
by what is known of the subject, they gave a definition some- 
what as follows: A porism is that which in the hypothesis is 
less complete than a local theorem. And loci are instances 
of this class of porisms, and they abound in analysis. But this 
class of questions, because it has a wider range than other classes, 
has been separated from porisms, and a collection of them has 
been made, and treatises written on them and handed down to us. 
And of these loci some are plane, some are solid, some are linear, 
and some depend on mean proportionals. 

Now it sometimes happens that porisms have enunciations 
which are contracted owing to the abbreviated form of expres- 
sion, and in them much is generally supposed to be supplied 
(by the reader), so that many geometers only partially understand 
the matter, and do not comprehend the more important part 
which is implied in them. And in the case of porisms it is not 
possible to include many in one proposition because Euclid him- 
self has not given many out of each class; but for the sake of 
example out of a great number he has given a few belonging to 
the same class at the beginning of his first book, all of them, 
about 10 in number, belonging to that somewhat numerous class 
of loci ; wherefore finding it possible to include these in one state- 
ment, we have given it as follows*: 

In Fig. 22, p. 36, given four straight lines BG, GA, AB, DE 

* For the Greek text of this Porism which is given by Pappus withou 
either figures or letters, see p. 121. 




pappus' account of EUCLID'S PORISMS 117 

intersecting by pairs in the points A, B, (7, P, Q, E, if three of the 
points P, Q, E lying on one of them BB (or two of them in the 
case of parallelism), [i.e. when £G is parallel to DE, in which case 
Pis at infinity], are fixed, and of the other three points two, viz. 
B and C, move along the fixed straight lines OJD, OE, the last 
point A will also move along a fixed straight line. 

This enunciation refers to only four straight lines, of which 
not more than two pass through the same point, and does not 
mention the fact that a similar proposition holds for any pro- 
posed number, which may be stated as follows : If any number of 
straight lines cut one another, of which not more than two pass 
through the same point, and it is also given that all the points 
of intersection lying on one of them are fixed, and of those which 
lie on the others each moves along a given straight line ; or more 
generally as follows: If any number of straight lines cut one 
another, of which not more than two pass through the same point, 
and it is also given that all the points of intersection lying on 
one of them are fixed, and if the number of the rest is a tri- 
angular number whose side is the same as the number of the 
fixed collinear points, and no three of them {i.e. of the points of 
intersection of the moving lines with the given lines) are at the 
vertices of a triangle, then each of the remaining points describes 
a straight line. 

Now it is not likely that the writer of the Elements was not 
aware of this, but he was merely stating the first principles, and 
in the case of each of the porisms he seems to have put forth 
only the first principles and germs of many important properties, 
and of these the classes are to be distinguished not by the differ- 
ences of their hypotheses, but by those of their conclusions and 
qusesita*. For all hypotheses, being very special in character, 

* Simson {Rel. Op. p. 349) explains this to mean that there are many 
porisms which have different hypotheses, but in all of which the conclusion 
is * that a certain point lies on a fixed straight line ' or ' that a certain 
straight line passes through a fixed point,' &c. 



118 CROSS-RATIO GEOMETRY [APP. I 

differ from each other, and each conclusion and qusesitum, although 
one and the same, are to be considered separately in the many 
different hypotheses. Noav in the first book the following classes 
are to be formed by the qusesita in the propositions. There is a 
diagram referring to this at the beginning of the seventh book*. 



[Classes of Porisms.] 

I. If from two given points (two) straight lines are drawn 
intersecting on a given straight line, and if one of them cuts off 
from a given straight line (a segment measured) from a given 
point in it, the other will also cut off from another straight line 
(a segment measured from a given point in it) having a given 
ratio (to the former segment) t. 

II. A certain point lies on a given straight line. 

III. The ratio of a certain line to a certain other line 
is given. 

TV. The ratio of a certain line to a segment (is given). 

V. A certain line is given in position. 

YI. A certain line passes through a given point. 

VII. The ratio of a certain line to a segment between a 
certain point and another given point (is given). 

VIII. The ratio of a certain line to a segment drawn from a 
certain point (is given). 

IX. The ratio of a certain rectangle to that contained by 
a given line and a certain other line (is given). 

X. Of a certain rectangle one part is given, and thei 
remainder has a given ratio to a segment of a line. \ 

XI. A certain rectangle, or a certain rectangle together with 
another given rectangle is (given), and the former has a (given) i 
ratio to a segment of a line. 

* This diagram is unfortunately missing, 
t For proof see Simson, de Porismatibus, p. 400, Prop. 23, and Chasles^ 
Porismes d'Euclide, p. 114, Prop. 11. 



CLASSES OF PORISMS 119 

XII. A certain line together with another line with which 
a certain other line is in a given ratio is itself in a (given) ratio 
to a segment drawn from a certain point to a given point. 

XIII. (A triangle whose vertex is) at a given point, and 
(whose base is) a certain straight line is equal to (a triangle whose 
vertex is) at a given point, and (whose base is a segment drawn) 
from a certain point to a given point. 

XIV. The ratio of the sum of a certain pair of lines to 
a segment drawn from a certain point to a given point (is given). 

XV. A certain line cuts off from two given lines (segments) 
which have a (given) rectangle. 



In the second book the hypotheses are different, but the 
greater number of the qusesita are the same as those in the first 
book with these additional ones. 

XVI. A certain rectangle or a certain rectangle together 
with a given rectangle has a (given) ratio to a segment of a line. 

XVII. The ratio of the (rectangle contained) by certain 
lines to a segment of a line (is given). 

XVIII. The ratio of the rectangle, one of whose sides is the 
sum of a certain pair of lines and the other side the sum of 
a certain other pair of lines, to a segment of a line (is given). 

XIX. The rectangle, one of whose sides is a certain line 
and the other side the sum of a certain line and of another one 
to which a certain line bears a given ratio, and the rectangle 
whose sides are a certain line and another line to which a 
certain line bears a given ratio have their sum in a (given) 
ratio to a segment of a line. 

XX. The ratio of the sum of two rectangles to a segment 
drawn from a certain point to a given point (is given). 

XXI. The rectangle contained by a certain pair of lines 
is given. 



120 CROSS-RATIO GEOMETRY [aPP. I 

In the third book the majority of the hypotheses relate to 
semicircles, and a few to the circle and segments. Most of the 
qusesita are similar to those given above, with these additional 
ones. 

XXII. The ratio of the rectangle contained by a certain 
pair of lines to that contained by a certain other pair of lines 
(is given). 

XXIII. The ratio of the square on a certain line to a segment 
(is given). 

XXIV. The rectangle contained by a certain pair of lines 
(is equal) to the rectangle contained by a given line and a line 
drawn from a certain point to a given point. 

XXV. The square on a certain line (is equal) to the rectangle 
contained by a given line and an abscissa of a line between a 
given point on it and the foot of a perpendicular. 

XXVI. The sum of a certain line and of a line to which 
a certain other line has a given ratio, has a (given) ratio to a 
segment. 

XXVII. There exists a certain given point from which 
straight lines drawn to certain (circles) will enclose a triangle of 
given species. 

XXVIII. There exists a certain given point from which 
straight lines drawn to a certain (circle) cut off equal arcs. 

XXIX. A certain straight line is either parallel to, or 
makes a given angle with a straight line drawn to a given point. 

The three books of porisms have 38 lemmas, and they contain 
171 theorems." 

Poncelet in the introduction to his Traite de proprietes 
projectives des figures (1822) suggested that the porisms were 
projective properties deduced by Euclid from considerations of 
perspective, but Chasles has made it pretty clear that Euclid's 
treatise was concerned with the principles of cross-ratios. Book i 
dealing with homographic divisions on two lines. Book ii with 



I 



simson's discovery 121 

co-axial ranges, and Book iii with the anharmonic properties of 
the circle. 

In order that the student may understand the nature of the 
difficulty which presented itself to the geometers of the 17th, 
18th, and 19th centuries, several of whom attacked the question, 
we give the Greek text of the only Porism enunciated by Pappus. 

€av VTTTLOv 7j TTapvTTTLOV Tpitt TO, CTTt /xttts (TrffxcLa [tj TTapaWrjXov 
€T€pa TO, Svo] 8c8o/u.€va y, to. 8c Xolttol ttXtjv €i/o? aTTTTjTaL Oea€L ScSo- 

A paraphrase of the above general proposition is given on 
p. 116, with letters and a reference to a figure. Simson explains 
the term vtttlov to mean a quadrilateral in which two adjacent 
sides tend to meet in a direction opposite to that in which the 
others tend to meet, whilst irapv-n-TLov is a quadrilateral in which 
two adjacent sides tend to meet in the same direction as the 
others, e.g. in Fig. 22, p. 36, ABOC is an example of vittlov, 
since BO, CO tend to meet in the opposite direction to BA, CA, 
and PQAC, PBAR are examples of Trapv-n-TLov, for PQ, AQ tend 
to the same direction as PC, AC, and similarly PB, AB tend to 
the same direction as PE, AR. 

The whole subject was an enigma to Fermat (1601 — 1665), 
and even Halley (1706) confesses "Porismatum descriptio nee 
mihi intellecta, nee lectori profutura. Quid sibi velit Pappus 
baud mihi datum est conjicere." It was not until 1723 that the 
key to the mystery was found by R. Simson (1687 — 1768), pro- 
fessor of mathematics at Glasgow, and his account of his discovery 
will always be read with interest. "I often tried," he said, "but 
always in vain, to understand and restore the only porism which 
survives out of all that were in the three books, and as my medi- 
tations on it took up too much of my time I determined that I 
would never touch the subject again, especially as Halley had 
given up all hope of understanding it. Consequently, whenever 
it occurred to me, I always refused to dwell upon it. However, 
sometime afterwards it presented itself to my mind when I was 



122 CROSS-RATIO GEOMETRY [APP. 1 

off my guard, and had in fact forgotten all about it, and it held 
possession of my thoughts until at length a glimmer of light was 
thrown upon it which gave me hopes of discovering Pappus' 
general proposition, and this, after much thought, I was at 
length enabled to restore." 

In his Opera qucadam reliqua published in 1776, eight years 
after his death, we find a restoration of Euclid's treatise con- 
taining 93 propositions*. This roused a fresh interest in the 
subject, and in 1860 Chasles published his restoration "Conforme- 
ment au sentiment de R. Simson sur la forme des enonces de ces 
propositions." For further information we must refer the student 
to Chasles' work Les trois livres de Porismes d'Euclide, 1860, 
324 pp. in which he gives 219 Porisms, with a complete historical 
account of Euclid's treatise, and demonstrates its hiomographical 
character. 



The following are Props. 127 — 144 of the 7th book of Pappus. 
The enunciations are the literal translations of the Greek text, 
but I have substituted proofs which are intended to shew the 
connection of the propositions with the theory of cross-ratio. 



LEMMAS ON EUCLID'S PORISMS. 

On the 1st Porism of Bk I. 

I. In Fig. 60 let ac : cb' = ah : he, and let CD he joined. I say 
that the straight lines ac', CD are parallel. 

* On the outside cover of an Appendix (1847) to Potts' larger edition of 
Euclid there was a notice that it was proposed to publish by subscription a 
translation of Simson's Restoration of the Porisms. The translation was to 
be preceded by a discussion of their peculiar character, together with a full 
development of the algebraical method of investigating them. 

If a number of subscribers had been obtained sufficient to defray expenses, 
it was intended to print the work at the University Press in octavo, and to 
issue it at a price not exceeding ten shillings. 



I 




LEMMAS ON EUCLID S PORISMS 



123 



[Consider ah as a transversal of the quadrilateral ABCD. 
The given relation is equivalent to 

ac : ah' = ah : ac', or ac . ac' = ah . ah'. 




Fig. 60 (i, II, IV). 

Therefore a is the centre of the involution system of points in 
which the transversal meets the sides and diagonals of the quad- 
rilateral. 

Therefore a, the conjugate of «, is at infinity, and the trans- 
versal is parallel to CD. See Arts. 98, 119.] 



On the 2nd Porism. 

II. /n the same figure let ah he parallel to CD, and let 
be : ha = ch' : ca, I say that the points A, B, a are collinear. 

[The relation is equivalent to 

ac' : ah = ah' : ac, or ah . ah' = ac . ac' ; 
therefore the conjugate of the point at infinity is a; i.e. the 
transversal meets the side AB in the point a. Art. 119.] 

III. T2V0 straight lines ad, ad' are drawn cutting the three 
straight lines Oh, Oc, Od. I say that 

ah' . d'c : ad' . ch' = ah .dead . he. 
[i.e. {ac'h'd') = (achd). Art. 21.] 



124 



CROSS-RATIO GEOMETRY 



[APP. I 




Fig. 61 (ill, X, XI, XIV, xvi). 

IV. In Fig. 60 let aa . be : ac . ha! = aa' . b'c : ab' . ca'. I say 
that Cj B, a' are collinear. 

[Otherwise, a transversal meets two diagonals of a quad- 
rilateral in c, c', two opposite sides in 6, b' and a third side in a. 
If a point a' is taken on the transversal such that {a'cah) = {a'b'ac') 
the transversal will meet the fourth side in the point a. Art. 
119.] 




D r c F 

Fig. 62 (v, vi). 

V. In Fig. 62 {it is proved elsewhere that) DF : FC = Dr: rC. 
Therefore if DF \FC = Dr\ rC, I say that D, G, B are collinear. 

VI. In Fig. 62 if AB and DC are parallel, (it is proved else- 
where that) Dr = rC. Suppose now that Dr = rC, I say that A Band 
DC are parallel. 

[DrCF is a harmonic range, and if Dr = rC, F is at infinity, 
i.e. AB is parallel to DC. Arts. 118, 29.] 



LEMMAS ON EUCLID S PORISMS 
B 



125 




c h 

Fig. 63 (vii). 

VII. Let Fc he a mean proportional between c'h and c'b'. 
I say that A C and Fc' are parallel. 

[Fc'bb' is a transversal of the quadrilateral ABCD. F is one 
of the double points of the involution, and since c'b . c'b' = c'F'^, c' 
is the centre, and its conjugate c, the intersection oi AG and Fc , 
is at infinity.] 

/ 




Fig. 64 (viii). 

VIII. Let abcdefg be a Bomiscus [i.e. a figure like a little 
altar with unequal sides], and let de be parallel to be, and eg to bf. 
I say thatfd is parallel to eg. 

[Otherwise, if bcgedf is a hexagon inscribed in the line-pair 
bdg, cefi and if the opposite sides ge, fb are parallel, and also ed 
parallel to 6c, then shall the remaining pair eg and^o? be parallel, 
i.e. the Pascal line is altogether at infinity. See Art. 51.] 

IX. Pth' is a triangle, and in it are drawn Pr, Pk' , and AD 
is d7'awn parallel to th', and let A F, DF be drawn to a point F on 
th' such that tF : Fh' = rF : Fk'. I say that BG is parallel to th'. 

[By Art. 120 Pt, Pr, Pk', Ph' are four rays of an involution 



126 CROSS-RATIO GEOMETRY [APR I 

pencil drawn from P to the four vertices of the quadrilateral 
ABCD. 




r F k' 

Fig. 65 (ix). 

Therefore (t, k') and (r, h') are pairs of conjugate points. 
Also since Ft . Fk' = Fr . Fh\ F is the centre of the involution, 
and therefore its conjugate is at infinity; therefore the sides 
AD^ £C meet at infinity, i.e. those sides are parallel.] 

X. In Fig. 61, from the point a two straight lines abc^ ah'c' 
are drawn cutting the two lines Obb', Occ', and let d, d' he points 
on them such that ca .bd:cd.ba = ac' . d'b' :ab' . d'c. I say that 
the points 0, d, d' are collinear. 

[Since {chad) = (c'b'ad'), .'. 0, d, d! are collinear by Art. 23.] 

XI. In the same Fig. 61, let Ob'c' be a triangle., and OD 
parallel to b'c ^ and draw Da meeting b'c' in a. I say that 

aD .bc-.ab.cD — c'b' : b'a. 

[In X, d is at D, and d' at infinity, .'. {acbD) = (ac'b' ao ) =—rr, •] 

c 

XII. Having proved the preceding propositions we will now 
shew that in Fig. 66 i/* ac', Oe are parallel, and are intersected by 
the straight lines Oc, c'd, da, ae, and the lines Ob' , b'e are drawuy 
I say that the points k, m, b are collinear. 

[This is Pascal's Theorem for the hexagon Ob'eadc inscribed 
in the line-pair Ode, ah'c, these lines being parallel. Art. 51. 



LEMMAS ON EUCLID'S PORISMS 



127 




Fig, 66 (xii, XV, xviii). 



Since Ob, Oc, Od are intersected by abed, ab'c', of which the 
latter is parallel to Od, .'.by Lemma XI, {acbd) = -rr, . 

Similarly, since ed, ea, eb' are intersected by c'd, c'a, of which 
the latter is parallel to ed, 

.'. (c'lkd) = ^ . .-. by Art. 3, (acbd) = (Ic'kd). 

And al, cc' meet in m. .'.by Art. 23, b, m, k are collinear, bmk 
being the cross-axis. Art, 50.] 



128 



CROSS-RATIO GEOMETRY 



[APP. I 




Fig. 67 (XIII, xix). 

XIII. Now let ac\ Oe he not parallel, and let them meet in 
the point n. I say that in this case also the points k, m, b are 
collinear. 

[This is Pascal's Theorem for the hexagon Ob'eadc inscribed 
in the line-pair Ode, ab'c' in the general case. Art. 51. 

Since On, Ob', Oc are cut by ad, ac, 

.'. (acbd) — {acb'n) by Lemma III 

= (Ickd), since en, ea, eb' are cut by c'd, c'n. 

And al, cc meet in m, .'. by Art. 23, b, m, k are collinear and 
bmk is the cross-axis.] 

XIV. In Fig. 61, let OD be parallel to ac, and let Ob', aD be 
drawn, and onhD take a point c such that c'b' : b'a = aD . be : cD . ab. 
I say that the points 0, c, c' are collinear. 

[See Lemma XI (acbD) = {ac'b'oo ). Then Art. 23.] 

XY. The previous propositions hamng been proved, in Fig. 66 
let ac' be parallel to Oe, and let them be cut by the straight lines c'd, 
da, eb', b'O, and draw the lines ae, bk. I say that the points c', m, 
are collinear. 



LEMMAS ON EUCLID's PORISMS 129 

[Otherwise, 0, o?, e and a, 6', c' are two triads of points on 
two parallel straight lines. Oh\ ad meet in 6, c'd, h'e meet in A;, 
and ae, hk meet in m. Then 0, m, c' are coUinear. The converse of 
Lemma XII.] 

XVI. The same as X. 

XVII. The same as XV, except that Oe, ac' are not parallel. 
[The converse of XIII.] 

XVIII. In Fig. 66 Oh'c' is a triangle, and Od is drawn 
parallel to h'c', and da, eg are drawn so that ab'^ : ac' . c'h' — h'g : gc'. 
I say that if h'd is drawn, the points h, h', c' are collinear. 

rmi • 1 ^- • • 1 ^ X %-«c' ah' dc.ha , 
[The given relation is equivalent to ,~ r, = }J~>= 3 i "^y 

XI, i.e. (agc'b') - (acbd). And since eg, h'd meet in k', .'. h, k', c' 
are collinear by Art. 23.] 

XIX. In Fig. 67 from the point c' are drawn two straight 
lines c'd, c'n cutting the three straight lines, en, ea, eh', and let 
c'd : dl = kc' : kl. I say that c'n : na = c'h' : h'a. 

[The harmonic pencil e (c'kld) determines a harmonic range on 
any transversal c'h' an by Art. 31.] 



M. 



CHAPTER XI 

ANHARMONIC PROPERTIES OF POINTS AND TANGENTS OF A 
CONIC. THE LOCUS AD TRES ET QUATUOR LINE AS 

126. We will now proceed to apply the principles of homo- 
graphy to conies, and in doing so we shall assume that the 
student possesses a knowledge of the elementary properties of 
conies as given in the ordinary text-books, and we will first give 
two propositions which are, as it were, the foundations on which 
we shall build. The method we have adopted is due to B. W. 
Home, Quarterly Journal of Mathematics, Vol. iv, 278, 1861. For 
other ways of opening the subject see Chasles, Traite des Sections 
Coniques (1865). See also his Apergu Historique, Notes xv, xvi, 
and his Traite de Geometrie Superieure, Chap. xxv. 

127. A and B are two fixed points on a conic, focus S, and 
is any variable point on the curve. OA and OB meet the S- 
directrix in a, h. Then the angle aSb = ^A SB = const. 

By the focus and directrix definition of a conic, 

SO:SA = Oa: Aa, 

.'. Sa bisects the angle ASO\ .'. aSO' = ^ASO'. 

Similarly, SO:SB = Ob : Bb. 

/. Sb bisects the Single BSO\ .'. bSO' = ^BSO'. 

.'. aSb = bSO' - aSO' = \ {BSO' - ASO') = IASB. 



126-128] 



TWO PROPERTIES OF CONICS 



131 




Fig. 68. 

128. If the, tangents at A, B meet the tangent at any point 
in a, /3, the angle aS^= ^ AS B = const. 

Since tangents to a conic subtend equal angles at a focus, the 
angle 

aSO = aSA = ^ASO, 

and ftS0=/3SB = iBS0, 

:. aSf3 = iASB = aSb. 

Dep. The pencil formed by joining four points on a conic 
to a fifth point on the curve is called a conic- 



Conic-pencil 



pencil *. 



Prof. A. Lodge, 1908. 



9—2 



132 CROSS-RATIO GEOMETRY [CH. XI 

129. If A, B, C, D are four fixed points on a conic, and 
any variable point on the curve, the conic-pencil {ABCD) has a 
constant cross-ratio *. 

In Fig. 68 {ABCD) = {abed) 

= S (abed) hj Art. 24, 
and by Art. 127 the angles subtended at S by pairs of the points 
a, b, c, d are half the angles subtended at S by the pairs of 
corresponding points A, B, C, D, and therefore the pencil S {abed) 
is constant. Hence the conic-pencil {ABCD) is constant. 

As the cross-ratio of the conic-pencil {ABCD) is the same 
for all points on the conic, we may speak of it as the cross- 
ratio of the four points A, B, C, D, and the above property may 
be stated : 

The cross-ratio of four fixed points on a conic is constant, 
meaning that the cross-ratio of the conic-pencil formed by joining 
the four points to any variable point on the curve is constant. 

From the above it follows that if 0, 0' are two positions of 
the variable point, the conic-pencil {ABCD) = 0' {ABCD), and 
this fact is independent of the position of the points A, B, C, D. 
Hence we may now suppose 0, 0' to be two fixed points, and 
A, B, C, D to be any four positions of a variable point on the 
conic, and we have the theorem : 

If two fixed points on a conic are joined to a variable point on 
the curve, the pencils so formed are homographic, since any four 
positions of the variable point give two pencils, centres 0, 0', having 
the same cross-ratio. 

Hence, considering any six points on a conic, 

(a) If we fix four of the points, the other two give us a pair of 
equicross pencils, and the conic-pencils formed by drawing rays 
to the four points from any variable point on the curve are equi- 
cross. 

* Chasles, 1829; Steiner, 1832. 



129-130] CONIC-PENCILS AND FIXED TANGENTS 133 

(p) If we fix two of the points and suppose the others variable, 
the pencils formed by drawing from the two points rays inter- 
secting on the conic are homographic. 

130. If the tangents at four fixed points A, B, C, D on a 
conic meet the tangent at any variable 2>oint in a, /8, y, 8, the 
range (a^yS) has a constant cross-ratio. 

In Fig. 68 by Art. 128 the pencils S{ahcd) and S (afSyS) are 
superposable. .'. S (a/3y8) = >S' (abed) = (abed) ^ (ABCD). 

Therefore (a^yS) = const. 

As the cross-ratio of the range {a(3yB) is the same for the 
tangent at any point of the conic, it may be called the cross- 
ratio of the four tangents at A, B, C, D, and the above property 
may be stated : 

The cross-ratio of four fixed ta^ngents is constant, meaning that 
the cross-ratio of the range hi which four fixed tangents cut any 
variable tangent is constant. 

As in the previous article, we might take two positions TP, 
TQ of the variable tangent, and let the four tangents atA,B,C,D 
meet them in a, /3, y, 8 and a, fS', y', 8'. Then by the above, 
(a/3yS) = {a'jS'y'S'). This fact is quite independent of the positions 
of the points A, B, C, D. Hence we may consider TP, TQ as 
two fixed tangents, and the tangents at A, B, (7, D as any four 
positions of a variable tangent. This gives us the theorem : 

If two fixed tangents are cut by a variable tangent the ranges 
80 formed are homographic, 

since any four positions of the variable tangent give points on the 
fixed tangents which have the same cross-ratio. Hence, consider- 
ing any six tangents to a conic, 

(a) If we fix four of the tangents, the other two give us 
a pair of equicross ranges ; and so the ranges formed by the 
intersections of the four fixed tangents with any variable tangent 
are equicross. 



184 CROSS-RATIO GEOMETRY [CH. XI 

(^) If we fix two of the tangents, and suppose the others 
variable, we may remove the restriction to four, and think of any 
number, and these will determine homographic ranges on the two 
fixed tangents. 

Note. The properties in Arts. 129, 130, being projective, 
could of course be deduced from the corresponding properties of 
the circle. 

131. It follows from the preceding article that : 

The cross-ratio of the conic-pencil of any four points on a conic 
is equal to the cross-ratio of their tangents. 

This of course is a very abbreviated statement, and the 
student should realize that its meaning, expressed in full, is : 

The cross-ratio of the conic-pencil formed hy joining four fixed 
points A, B, C, D on a conic to any variable point on the curve is 
equal to the cross-ratio of the range formed hy the points of inter- 
section of the tangents at A, B, C, D with any variable tangent to 
the conic. 

132. The converses of Arts. 129, 130 are very important, 
and are of two distinct classes. In the first it is assumed that 
there is given a conic-pencil or a tangent range, and it is proved 
that equi-pencils or equi-ranges belong to the same conic, and 
therefore incidentally that only one conic exists in each case. 
The second class is more general, and does not assume the exist- 
ence of a conic, but only that of equi-pencils or equi-ranges, and 
proves that the locus or envelope is a conic, and the theorems of 
this latter class, which we may term complete converses, really 
include those of the former, which may be called partial con- 
verses. 

133. We will now prove the partial converse of Art. 129 (a). 
Ifive have given four fixed points on a conic, and if the pencil 

formed by joining them to a point P has the same cross-ratio as the 
conic-pencil formed hy the four points, the point P lies on the conic. 



131-134] CONIC-PENCILS AND FIXED TANGENTS 135 

Let A, £, C, D be the four fixed points, and suppose F is not 
on the conic. Let one of the rays FA meet the conic in F'. 
Then the pencil F{ABCD)= the conic-pencil F' (ABCD), and 
these have a common ray, and are therefore in perspective, 




Fig. 69. 

i.e. by Art. 25 the points B, C, D are collinear, which is contrary 
to the supposition that they are on the conic. 

CoR. If a conic can be drawn through five points, only one 
conic can be so drawn. 

134. Given four fixed tangents to a conicj if they form on any 
straight line the same cross-ratio as that formed by them on any 
fifth tangent^ the straight line will touch the conic. 

Partial converse of Art. 130 (a). See Fig. 70. 

Let the tangents at the fixed points A, B, C, D intersect the 
straight line L at a, b, c, d. Then if the line L is not a tangent,, 
from one of the points of intersection, as a, draw a tangent 
meeting the three other fixed tangents at 6', c', d'. 

Then the ranges {ab'c'd') and {abed) are equicross and have a 
common point a, and are therefore by Art. 23 in perspective, 
i.e. the three tangents bb', cc', dd' are concurrent, which is absurd. 



136 



CROSS-RATIO GEOMETRY 



[CH. XI 



Cor. If a conic can be drawn to touch five lines, only one 
conic can be so drawn. 




Fig. 70. 



Proofs of Arts. 129 {(3) and 130 {/3) based upon 
Propositions given by ApoUonius. 

135. In ApoUonius, Bk iii. Prop. 54, we find the following 
property. See Fig. 71. 

TIf TJ' are fixed tangents to a conic, P any variable point on 
the curve. Through I and J' lines are drawn parallel to the 
tangents, meeting J'F, IP in a, a'. Then for all positions of P the 
rectangle la . J' a is constant. 

Through P draw a line PP' parallel to IJ' meeting the conic 
in P', the tangents in A", K', and the diameter C^' in W. Let 



135] 



PROPOSITION OF APOLLONIUS 



137 



Ca, (7yS, Cy be the semi-diameters parallel to IJ\ TI, TJ'. To 
shorten the statement of the proof we shall assume a knowledge 
of the following properties : 

(1) KP .KF'.KP=Ca^:C^. 



P^ 



M. 



Fig. 71. 

(2) W bisects FP', and also KK', and consequently 

KP = K'F, KF = K'P, 
and therefore KP . KP' = K'P' . K'P = KP . K'P. 

(3) From the similar triangles alJ\ J' K'P, 

Ia:IJ' = K'J'.K'P, 
and from the similar triangles a J' I, IKP, 
J' a' : IJ' = KI : KP. 



138 



CROSS-RATIO GEOMETRY 



[CH. XI 



By(l) 

Similarly 



KF 
K'J"" 



Co? ~ KF 

9f^ 

Ca^ K'P . K'F 
Cc^ ~ KF ' K'F 



KF 



KF.K'F 
K'F^ 



by (2). 



KF . K'F 

la J' a' , .^. 



7„.^V = 54^./^': 



const. 



Hence by Art. 64 the ranges {a), (a) are homographic, the 
points /, J' corresponding to the points at infinity, and therefore 
the pencils I {a'), J' (a), i.e. the conic-pencils I{F), J' {P), are 
homographic. Consequently Apollonius' property may be stated : 

If /, J', two fixed points on a conic, are joined to any number 
of points on the curve, the conic-pencils so formed are homogr^aphic, 
which is equivalent to the theorem of Art. 129 (^). Also, obviously, 
the ray in the pencil J' (a) corresponding to the ray IJ' in the 
pencil I (a') is the ray drawn from J' to the point at infinity on 
la, i.e. the tangent at J'. Similarly the tangent at /corresponds 
to the ray J' I in the pencil J' (a). 

136. A variable tangent aa meets TP, TQ', two fixed tangents 
to a conic, in two ranges (a), (of) which are homographic. 

a" I Q T'_ 




136-137] CHASLES' THEOREMS 139 

PF, QQ' are the diameters through P, Q', and CD is the 
semidiameter conjugate to ^^'. The tangent aa' meets the tangent 
at Q in a". It is proved in Apollonius, Bk iii, Prop. 42, that 

QI.Q'T=^CD''=Qa" .Q'a' (A). 

Assuming this result (A), for proof of which see Milne and 
Davis' Geom. Con. Art. 134, or any other text-book, and the 
truth of which can be seen at once by orthogonal projection from 
a circle, we have 

QI:Q'a' = Qa":Q'T 

= Qa"-QI'.Q'T-Q'a\ 
:. J'Q' '.Q'a' = Ia":a'T, for QI=J'Q' 

= la : aT, 
.'. J'Q':J'a'=Ia:IT, 
.'. la. J'a= IT. rQ' = const* 
Hence the ranges (a), (a) are homographic, and we have 
the theorem of Art. 130 (^). 

Anharmonic Properties of Points and Tangents 
of a Conic. 

137. We will now give the complete converses of Arts. 1 29 (/3), 
130 (^), which are due to Chasles, and are two of the most 
important propositions in this part of the subject. 

(1) Given tivo homographic pencils not in perspective, the 
intersections of corresponding rays lie on a conic which passes 
through the centres of the pencils. 

(2) Given two homographic ranges not in perspective on two 
given straight lines, the lines joinifig pairs of corresponding points 
envelop a conic which touches the two given lines. 

Chasles called (1) the ayiharmonic property of the points of a 
conic, and (2) the anharmonic property of the tangents of a conic, 
and these, with their converses given in Arts. 129, 130, he takes 

* Newton's Principia, Bk i, Sect, v, Lemma 25. 



140 



CROSS-RATIO GEOMETRY 



[CH. XI 



as the fundamental propositions on which he bases his Traite des 
Sections Coniques (1865). They are first met with in Notes xv, 
XVI of his Aper^u Historique (1837), where he proves the 
properties for the circle, and then employs the property of 
Art. 21, which shews that the cross-ratio of four collinear points 
is unaltered by projection. In his Geometrie Superieure (1852), 
Chap. XXV, he treats independently the locus and envelope 
referred to, and shews that the curve locus 

(1) passes through the centres of the pencils, 

(2) cannot meet a straight line in more than two points, 

(3) if about two of its points rays are rotated intersecting 
on the curve, the rays form two homographic pencils, 

(4) two such curves can be considered as . homographic 
figures, and can be placed in perspective with each other, and can 
therefore always be considered as the plane section of a cone on a 
circular base, with similar properties for the curve envelope, and 
in this way he deduces that both curves are conies. 

The proofs given below are taken from Chasles' Traite des 
Sections Coniques, Arts. 8 and 9, where he adds the note "L'idee 
de construire sur la figure meme le cercle dont la courbe en- 
gendree sera la perspective, m'a ete suggeree par M. J. Delbalat." 

138. Given two homographic pencils not in perspective, the 
intersections of corresponding rays lie on a conic which passes 
through the centres of the pencils. 

A 




8 y fi^ 

Fig. 73. 



138] CHASLES' THEOREMS 141 

Let P{ABG ...), Q(ABC ...) be two homographic pencils. 

It is required to shew that the locus of the points A, B, C ... 
is a conic passing through the points P, Q. 

Let FT be the ray in the first pencil corresponding to QP in 
the second, so that P {ABCT) = Q{ABCP). Describe a circle 
touching PT at P, and let it cut PA, PB, PC .., PQ in a, b,c ... q. 
Join qa, qb, qc ..., and produce them to meet the rays QA, QB, 
QC ... in a, p, y .... 

Now Q (ABCP) = P{ABCT) 

= P(abcT) 

= q (abcP)j 

since the pencils, centres P and q, are equiangular. 

And the pencils Q (A BOP), q (abcP) being equicross, and 
having a common ray QqP, are in perspective. Therefore a, ^, y, 
the points where corresponding rays meet, are collinear. Let L 
denote the line on which they meet. Then in the two triangles 
QAB, qab the lines joining the vertices Aa, Bb, Qq meet in P, 
therefore the triangles being co-polar are also co-axial, Art. 26. 
Therefore the sides AB, ab meet on the line joining a^, i.e. the 
line L ', i.e. the sides of the triangles QAB, qab intersect respec- 
tively in the line L', and this will still hold when the circle is 
rotated about L into any other position. Therefore in the new 
position QA, qa; QB, qb ; AB, ab are co-planar, and so Aa, Bby 
Qq meet in a point, the intersection of the three planes. Thus 
Bb, the line joining any two corresponding points, passes through 
a tixed point (not shewn in the figure), viz. that in which Qq 
intersects Aa, i.e. the two figures are in perspective, and 0, the 
centre of perspective, is the vertex of a cone passing through 
the circle and through the curve which is the locus of the 

points A, B, C Therefore this curve, being the section of 

a cone by a plane, is a conic. 



142 



CROSS-RATIO GEOMETRY 



[CH. XI 



139. Given tivo homographic ranges not in perspective on two 
given straight liyies, the lines joining pairs of corresponding points 
envelop a conic which touches the two given lines. 




Let (a6c ...), {a'h'c' ...) be two homographic ranges on the lines 
P, P'. It is required to shew that the lines aa, hh\ cc' ... envelop 
a conic touching the lines P, P. 

Let the lines P, P' intersect in the point p' , where p' is a 
point in the range {ah'd ...), and let jt? be its corresponding point 
in the range {ahc . . .). Describe a circle touching the line P at jo, 
and to it draw the tangents aa, 6^, cy . . . meeting the tangent p'lr 
in a, /3, y .... 

Then the range {ah' dp') = (abcp) by hypothesis 

= (a^yp') by Art. 130. 

The ranges (a'b'c'p'), (a^y/>') being equicross and having a 
common point p' are in perspective. Therefore the lines a'a, 6'/3, 
cy are concurrent. 



139-140] CHASLES' THEOREMS 143 

Now rotate the circle with its tangent pV about the line P. 
For any position of the moving plane the lines a' a, b'0, c'y are 
concurrent, since the ranges are still in perspective, p' being the 
common point. Let them meet in 0. Then with as vertex of 
projection the lines aa, bb', cc ... will be the projections of the 
lines aa, bjS, cy .... Therefore the curve enveloped by the lines 
aa', bb', cc' ... will be the perspective of the curve enveloped by 

the lines aa, bp, cy But the latter curve is a circle. Hence 

the former is a plane section of a cone on a circular base, i.e. a 
conic. 

140. A conic can be drawn through five points, no three of 
which are collinear. 

In Fig. 73 let P, Q, A, B, C be the five points. It is required 
to shew that they lie on a conic. 

Take P(ABC) and Q {ABC) as the characteristics of two 
homographic pencils, and draw PT in the first pencil correspond- 
ing to QP in the second, by Art. 58, so that 

P{ABCT) = Q{ABCP). 

Then with the same construction and demonstration as in 
Art. 138 it follows that the five points P, Q, A, B, C are the 
projections from of the five concyclic points p, q, a, b, c. There- 
fore the five points P, Q, A, B, C lie on the projection of a circle, 
i.e. on a conic. 

CoR. 1. Only one conic can be drawn through five given 
points. 

Cor. 2. By taking any other point d on the circle in its 
original position in Fig. 73, producing dq to meet L in 8, and 
finding the point D where Pd intersects 8Q, we can find as many 
more points as we please on the conic. Also, PT is obviously a 
tangent to the conic at P. 

To obtain the tangent at Q, draw the tangent to the circle at 
q, meeting L in K (not shewn in the figure). Then KQ is the 
tangent required. 



144 CROSS-RATIO GEOMETRY [CH. XI 

Since the tangents at any two corresponding points such as 
i), d intersect on Z, this provides us with a simple method of 
drawing the tangent at any point. 

The problem " to construct a conic through five points " is 
solved for an ellipse by Pappus, Bk viii, Props. 13, 14, where the 
method depends on a property given by Apollonius, Bk iii, 
Props. 16 — 23, sometimes called Newton's Theorem, viz, "The 
ratio of the rectangles contained by the segments of two inter- 
secting chords of a conic is equal to the ratio of the rectangles 
contained by the segments of any other pair of chords parallel to 
them." It is needless to say that the construction is theoretical 
rather than practical. 

For other solutions of the same problem see Newton's 
Principia, Bk i. Sect, v, Prop. 22, and Problem Lix of his 
Universal Arithmetic, and Art. 278 infra. 

141. It only remains to consider the case when three of the 
points, as A, B, C\ lie on a straight line. Let this meet the line 
through P, Q in K. The ray in the first pencil corresponding to 
QP in the second is now PQ, and if we describe a circle touching 
this ray at P, we should not obtain any point q, and the con- 
struction fails. 

It should be noticed that when this case occurs, the locus 
consists of the two lines through A, B, C and P, Q, because the 
common rays PQ, QP intersect each other not only at the point 
K, but anywhere along the line PQ. This is evidently the case 
of two pencils in perspective. 

142. A conic can he drawn touching five straight lines, no 
three of which are concurrent. 

In Fig. 74 let P, P', aa', bb', cc' be the five lines. It is 
required to shew that they are tangents to a conic. 

Take (abc') and {a'b'c') as the characteristics of two homo- 
graphic ranges, and let the point ^ on P correspond to p' the 



141-144] CONIC TOUCHING FIVE LINES 145 

intersection of the bases considered as a point on F, so that 
(abcp) = (ah'c'p). Then with the same construction and demon- 
stration as in Art. 139 it follows that the five lines P, F, aa\ hh\ 
ec' are the projections from of the five lines P, pV, aa, bjS, cy, 
which are tangents to a circle. Therefore the five given lines are 
tangents to the projection from of the circle, i.e. a conic. 

As in the previous article, if three of the lines aa, bb\ cc' 
meet in a point u, then if P, F meet in p', the envelope of the 
tangents degenerates into the two points it and p'. 

143. In questions relating to the anharmonic property of 
tangents of a conic, if the homographic ranges are of the second 
order, so that the points /, J' are at a finite distance, the conic 
which is enveloped by the lines joining pairs of corresponding 
points is a central one, the centre being the mid-point of the line 
joining the points /, J'. 

If the ranges are similar, the points /, J' are at infinity, so 
that the conic has its centre at infinity, and is therefore a 
parabola. 

If the points /, J' coincide at C, the intersection of the 
ranges, by Art. 68 (1) the homographic equation is 

Cm . Cm = const (A), 

the points of contact are at infinity, the conic is a hyperbola, the 
bases of the ranges are asymptotes, and (A) tells us that any 
tangent forms with the asymptotes a triangle of constant area. 

144. In dealing with the anharmonic property of points of a 
conic it is important to notice that any transversal is cut by the 
pencils in two homographic ranges, the common points of which 
are evidently the points where the transversal cuts the conic. 

If the ranges on the transversal are similar, one of the common 
points is at infinity, and the transversal is parallel to an asymptote. 
If in addition to being similar the ranges are superposable, the 
common points are both at infinity, and the transversal is an 
asymptote. 

M. 10 



146 CROSS-RATIO GEOMETRY [CH. XI 

The tangents to the conic at the vertices of the pencils are the 
rays which correspond to the line joining the vertices which it 
must be remembered are points on the conic. 

If these rays are parallel, the line joining the vertices is a 
diameter, and its mid-point is the centre of the conic. 

If we move the pencil, vertex 0', parallel to itself so that 0' 
is made to coincide with 0, any transversal will be cut by the two 
pencils, common vertex 0, in two homographic ranges. Let e, / 
be their common points. Then Oe, Of being the comraon rays of 
the pencils in their new position give us the directions of the two 
pairs of parallel corresponding rays of the pencils in their original 
position, and therefore these are the directions of the points at 
infinity. 

If the common points are real and separate, Oe and Of are 
parallel to the asymptotes, and the curve is a hyperbola. 

If the common points are real and coincident at e, the curve is 
a parabola, and Oe, is parallel to its axis. 

If the common points are imaginary, the curve is an ellipse. 

Locus ad tres et quatnor lineas. 

145. We will conclude this chapter with a short account of 
the above locus, which in point of interest can compare with any 
of the problems known to the ancients, the history of which makes 
the study of mathematics such a fascinating subject. 

In the general introduction to his Conies Apollonius says 
"The third book contains many curious theorems which are useful 
in the synthesis of solid loci, and in discriminating between their 
different cases ; of which theorems the greater part and the most 
interesting are new, and the knowledge of these enabled me to 
construct completely the locus ad tres et quatuor lineas, which was 
not completed by Euclid, but only a small part of it, and that not 
satisfactorily, for it was not possible for its synthesis to be com- 
pleted without the knowledge of the properties which I have 
discovered." 



145] LOCUS AD TRES ET QUATUOR LINE AS 147 

From Pappus, Bk vii, c. 36, we learn that the enunciations of 
the two cases of the locus were as follows : 

I. If three straight lines are given in position^ and from a 
point straight lines are drawn to thenn meeting them at given 
angles, and if the ratio of the rectangle contained hy two of these 
lines to the square on the third is given, the point lies on a given 



II. If there are four given straight lines, and from, a point 
straight lines are drawn to them meeting them at given angles, and 
if the ratio of the rectangle contained hy two of these lines to the 
rectangle contained hy the other two is given, in this case also the 
point lies on a given conic. 

As Apollonius does not make any further reference to the 
locus, nor give a solution in so many words, it was taken for 
granted that his solution had been lost, and no further search 
seems to have been made for one in his Conies, fortunately for 
mathematics, as Ball (1888) in his SJiort History of Mathematics, 
p. 242, tells us that " the general theorem had baffled previous 
geometricians, and it was in the attempt to solve it that Descartes 
was led to the invention of analytical geometry" (1659). Subse- 
quently, in 1687, a geometrical solution was given by Newton 
(as he thought for the first time) in his Principia^ Bk i, Sect, v, 
Lemmas 17 — 19, where he takes first the case of a trapezium, 
and then of any quadrilateral, and employs Apollonius, Bk in, 
Props. 17, 19, 21, 23. But, as was pointed out by a writer in the 
Math. Gazette, No. 6, of October 1895, Apollonius, Bk in, Prop. 54, 
referred to in Art. 135, not only virtually contains the solution, 
but also gives us the value of the constant ratio in terms of the 
diameters of the conic parallel to the given lines in the case 
where the straight lines are drawn to them at right angles. 

Thus, in Fig. 71, through P draw lines PF, PF', KPK' parallel 
to TJ', TI, IJ', and PL, PM, PN perpendicular to them. 

10—2 



148 CROSS-RATIO GEOMETRY [CH. XI 

PF PF' 
~ FJ'' F'l 

_ PF^ PT 
~ PK' ' PK 

_PN PN 
~¥L'PM' 

which is the locus ad tres lineas when the angles are right angles. 

If instead of being at right angles PL, PM, PN make angles ^^ , 

^2, ^3 with TJ', TI, IJ\ the proposition obviously still holds, but 

^1 1 c l.^. 1. J. J.' • C/S . Cy sin Oi . sin Oo 

the value or the constant ratio is now — _, ., ' . ^— . 

Ca' sm^ Os 

The locus ad quatuor lineas at once follows by repeated 
applications of the above, as shewn in Milne and Davis' Conies, 
Art. 254 (1894), where the value of the constant ratio is given in 
terms of the angles and the diameters parallel to the given lines. 

Seeing that the locus ad quatuor lineas can be derived from 
the converse of the anharmonic property of the points of a 
conic, the anharmonic property, as we might have expected, can 
be readily deduced from the locus. For let ABC J) be a quadri- 
lateral inscribed in a conic, P, P' any two points on the curve, 
and draw Pa, P'a! perpendiculars on A B. Then expressing twice 
the area of the triangle PAB in the two equivalent forms 

Pa.AB = PA. PB sin A PB, &c., 
we have 

Pa .Py.AB.CD s m A PB. sin CPJ) p,,«^^,, ,,., 
Pft.A.BC.AI) = smBPC.sinDPA =^^'^^^^^ ^^ ^''' ^^> 

and 

P'a'.P'y'.AB.CD _ sin APB. sin CPD _ pr.r^pry. 
P'/3'.P'S\BC.AD sin BP'C. sin DP' A ^ V^^^^> 



145] LOCUS AD TRES ET QUATUOR LINEAS 149 

Now by the property of the locus ' JL = ^ ' p,^, . 

Therefore P {ABCD) = F {ABCD). 

The importance of the locus is obvious when we notice that 
its two cases expressed in trilinear and quadrilinear coordinates 
take the well-known forms a(i = K-^'f and ajS — K^ySj kj and k^ 

being 7= — ^:iy and ^ ' -,, , where Ca, &c. are the semi-diameters 
^ Ca . Go Ca . Co 

parallel to the given lines. 

We shall see in a later chapter that Desargues', Pascal's and 

other well-known theorems are immediate consequences of it. 



EXAMPLES. 

1. OL, OU are two given lines, A, B are fixed points on OL. On OL' 
G, D are fixed points, and m, m' variable points. Am and Bm' meet in P. 
Find the locus of P 

(1) when the segment mm' is of constant length, 

(2) when Cm : Dm' is a constant ratio, 

(3) when the product Cm . Dm' is constant. 




Fig. 75. 

(1) As the segment mm' moves along the line OL', the ranges (w) and 
(w'), being superposable, are homographic. 

Therefore the pencils A (m) and B {vi') are homographic, Art. 42, and the 
locus of P is a conic passing through A and B, Art. 138. 



150 



CROSS-RATIO GEOMETRY 



[CH. XI 



Since the ranges (m), (m') made by the pencils on the transversal OL' are 
superposable, the line OL' is an asymptote, Art. 144. 
The hyperbola will be rectangular if mm' = AB cos 0. 

(2) Here the ranges (m), (m') are similar, but not superposable. 
Therefore the locus of P is a hyperbola having one of its asymptotes 

parallel to 0L\ 

(3) By Arts. 65, 70 the ranges (m), {m') are homographic, and therefore the 
locus of P is a conic. To determine its species draw rays through A parallel 
to the rays of the pencil B. Then the conic will be a hyperbola, parabola or 
ellipse according as the two pencils whose common vertex is A have their 
common rays real and separate, real and coincident, or imaginary. 

2. Shew that the locus of the vertex of an isosceles triangle whose equal 
sides pass through fixed points and whose base lies on a fixed straight line is 
a rectangular hyperbola having its centre midway between the fixed points, 
and one asymptote parallel to the fixed line. 




A' AH' D 

Fig. 76. 



Let 0, 0' be the fixed points, L the fixed line, ABC, A'B'C any two 
positions of the triangle. 

Then the angle ^0^' = ^--4'=P-£' = P0'P'. 

Therefore the pencils {A) and 0' (B), being superposable, are homo- 
graphic, and the locus of (7 is a conic passing through and 0'. 

To find the ray of the pencil corresponding to the ray O'O in the pencil 
0', let 00' meet L in G. Draw OD perpendicular to L, and take DH=DG. 
Then OH is the ray required, for the angle OHG=0'GH. 

Similarly by drawing O'D' perpendicular to L we can find O'H', the ray 
corresponding to 00'. And OH, O'H', which are the tangents at 0, 0', are 
parallel, for each makes with L the angle which 00' makes with L. There- 
fore 00' is a diameter, and the mid-point of 00' is the centre of the conic. 

By the geometry of the figure the ranges (A) and (B) are obviously 
similar, but not superposable, therefore one asymptote is parallel to L ; and 
the parallel lines OD, O'D' are a pair of corresponding rays, and are therefore 
parallel to the other asymptote. Therefore' the hyperbola is rectangular. 



EXAMPLES 



151 



3. The points A, B are fixed, and a moving point P lies on a fixed line 
L in the same plane with AB. Prove that the locus of the orthocentre of the 
triangle PAB is a hyperbola, one of whose asymptotes is perpendicular to 
AB, and the other perpendicular to L. Also shew how to draw the tangents 
at A and B. 

P 




A B Q 

Fig. 77. 

Let P, P' be two positions of the moving point, and 0, 0' the orthocentres 
of the triangles PAB, P'AB. 

Then the angle OAO' = PBP', and OBO' = PAP'. 

Therefore the pencils A[0), B{P) are superposable, as are also P(0), 
A (P). And the pencils A (P), B (P) are homographic by Art. 45. Therefore 
A (0) and B{0) are also homographic by Art, 44; and the locus of is a 
conic passing through A , B. 

To find the asymptotes. When P is at infinity, AP and BP are parallel 
to L. Therefore the rays through A and B perpendicular to L correspond, 
and being parallel give the direction of an asymptote. When P is at Q the 
rays through A and B perpendicular io AB correspond, and give the direction 
of the other asymptote. 

To draw the tangents at A and B. Draw Ap perpendicular to AB 
meeting L in p. Join Bp, and draw AT perpendicular to Bp. 

Then AT is the tangent at A. Similarly we can draw the tangent at B. 

4. Prove that the tangents to a parabola meet two fixed tangents in 
points forming two similar ranges, Art. 143. Also prove that if a straight 
line cuts the sides of a triangle in the points L, M, N such that the ratio of 
the segments is constant, it will envelop a parabola touching the sides of the 
triangle. 

Let any tangent cut the sides of a given tangent triangle of a parabola in 
the points P, Q, R, and let it cut the tangent at infinity in oo . Then by 

PR QR . PP 
Poo • goo ' ^'^' QR ' 



Art. 130 {PQRoD ) is const. , i.e. 



Then use Art. 134. 



152 CROSS-RATIO GEOMETRY [CH. XI 

5. Prove that if the corner of a rectangular piece of paper is folded down 
so that the sum of the edges unfolded is constant, the crease will envelop a 
parabola. 

Let A be the corner of the rectangle ABCD, and let the crease cut AB in 
m and AD in tw'. Then wjB + m'D = const. .*. by Art. 68 (3) the ranges 
(m), (m') are similar. .'. &c., Arts. 136, 143. 

6. Given two sides AB, AC oi & triangle in position, and 

(1) the sum or difference of these sides is constant, or 

(2) h . AB + k . AC=l, where h, k, i are constant, 
shew that the envelope of the base is a parabola. 

By Art. 68, in each case the ranges (B) and (C) are similar. ,". &c., 
Arts. 136, 143. 

7. Given two sides of a triangle in position and its area, shew that the 
envelope of the base is a hyperbola. 

The SLYea = ^AB .AC .sin A. :. AB .AC is constant. .-. by Art. 66 (1) 
the ranges (B) and (C) are homographic. .'. &c., Art. 143. 

8. Given the base and the difference of the base angles of a triangle, the 
locus of the vertex is a hyperbola. 

Take G, C two positions of the vertex, AB the base. 
Then the angle CBA - GAB=C'BA - CAB. 
:. CBC' = GAC'. .: pencil A {C)=B {C). 

9. Given in position two sides of a triangle, if the base subtends a 
constant angle at a fixed point, shew that it envelops a conic touching the 
two sides. 

Let BC be the base, D the fixed point. The pencils D (B), B(C) are 
superposable. .-. the ranges [B), (C) are homographic, &c. 

10. A perpendicular is drawn to each of two fixed tangents to a parabola 
at the point where it is cut by a variable tangent. Prove that their point of 
intersection lies on a fixed straight line. 

Let the variable tangent meet the fixed tangents in m, m', and let the 
perpendiculars at m and m' meet in P. 

By Art. 143 the ranges [m) and (m') are similar. See Chap. VI, Ex. 10. 

11. Given the base and area of a triangle, shew that the locus of its 
orthocentre is a conic passing through the extremities of the base. 

Let AB be the base. The vertex is on a line parallel to AB. Let C, C 
be two positions of the vertex, P, P' the corresponding positions of the ortho- 
centre. Then PAB is the complement of CBA, and P'AB the complement of 
CBA. .: PAP'=CBC, .: the pencil A{P)=B (C). Similarly PBP' = CAC, 
.-. B{P) = A{C). And B{C) = A{C) by Art. 45, .-. B{P) = A{P). .'. &c., 
Art. 138. 



EXAMPLES 153 

12. The line PQ subtends a right angle at each of the fixed points A 
and B, and the point P lies on a fixed straight line. Prove that the locus of 
Q is a hyperbola passing through A and B, and having one of its asymptotes 
perpendicular to AB, and the other perpendicular to the fixed straight line. 

By Art. 45 A{P) = B (P). Also A{Q) = A (P), being superposable, and for 
a similar reason B {Q) = B (P). /. A {Q) = B (Q). .: &c., Art. 138. 

The rays through A and B perpendicular to AB correspond, and are 
parallel, and therefore give the direction of an asymptote, Art. 144. 

13. AB, AC, two fixed tangents to a central conic, are cut by a variable 
tangent at m, m', and the segment mm' is divided in a constant ratio at P. 
The locus of P is a hyperbola having its asymptotes parallel to A B, AC, and 
having double contact with the given conic. 

Through P draw lines parallel to the fixed tangents, meeting AB, AC in 
fx, fi'. Itet mP = \ . mm' . Then Afji.= {l-\) Am, AiJ.' = \. Am'. 
Then the relation between m, m' is, by Art. 130 (j8) 

a . Am . Am' + h . Am + c . Am' + d = 0. 
.'. by substitution the relation between /a, fx' is 

.Afji..A/M' + - — -.Afi + ^.Afi' + d = 0. 



.•. the ranges (fx), (/*') are homographic, and if oo, oo ' are points at 
infinity on AB, AC, the pencils oo' (/x), oo {/j.') are homographic, and are not 
in perspective, because the line joining their vertices is not a common ray, 
since the relation between jx, fx' is of the second order. 

.-. by Art. 138 the locus of P is a conic through oo , oo ', i.e. a hyperbola. 
Moving the pencils, as in Art. 144, the common rays are evidently parallel 
to the fixed tangents. Also, for two positions of the variable tangent, P is a 
point of contact. .". &c. 

When the given conic is a parabola, the relation between m, m', and 
.-. also between fx, fx' is of the first order, and the pencils oo ' (yu), oo (jx') are 
in perspective. Cf. Chap. VI, Ex. 10. See also Chap. VI, Ex. 25. 

14. ABCD is a rectangle having the side BC produced to E so that 
CE = BC. Points L and M are taken in CD, DA respectively such that 
CL : CD = AM: AD. Prove that the locus of the intersection of EL and BM 
is an ellipse with semi-axes CD, CB. 

15. AB is a fixed diameter of a circle, mm' a movable chord perpen- 
dicular to it. Am, Bm' meet in P. Shew that the locus of P is a rectangular 
hyperbola. 

16. If AB is a fixed chord of a circle, and CD a chord of constant 
length but variable position, find the locus of the intersection of the lines 
AD, BG. 



154 CROSS-RATIO GEOMETRY [CH. XI 

17. AL, AL' are two given lines, and B a fixed point. Any circle passing 
through A and B cuts the lines in the points m, m'. Shew that the envelope 
of mm' is a parabola touching AL and AL'. 

18. If A and B are two fixed points on a conic, and a variable tangent 
meets the tangents at ^, J5 in m, m', prove that the locus of the intersection 
of Am', Bm is another conic. 

19. Through a fixed point ^ on a conic two straight lines AI, AV are 
drawn, S and &' are two other fixed points and P a variable point all on the 
conic. PS, PS' meet AI, AI' in Q, Q' respectively. Shew that QQ' passes 
through a fixed point. 

20. BB' is the niinor axis of an ellipse, and BP, BQ any two perpen- 
dicular chords through B. Shew that BP, B'Q intersect on a fixed straight 
line. 

21. AB is the base of an isosceles triangle ABC. On AC or AC 
produced any length AP is taken ; and on BC or BG produced a length BQ 
is taken such that the rectangle contained by AP, BQ is equal to a constant. 
Shew that the locus of the intersection of AQ and BP is a conic. 

22. Two sides AB, AC oi a triangle are given in position, and the base 
BC passes through a fixed point. BP is perpendicular to AB, and CP 
perpendicular to ^ C. Shew that the locus of P is a conic. Use the method 
of Chap. VI, Ex. 10. 

23. ABPC is a parallelogram having its sides AB, AC given in position, 
and the diagonal BC passes through a fixed point. Shew that the locus of P 
is a conic. 

24. Two sides of a triangle are given in position, and the circumcentre 
lies on a fixed straight line. Shew that the base envelops a parabola. 

25. Two tangents to a conic are fixed, and two others are drawn so as 
to form with the first pair a quadrilateral having two opposite sides along 
the fixed tangents. Shew that the locus of the intersection of the diagonals 
of this quadrilateral is a straight line. 

26. If the three sides QR, RP, PQ of a movable triangle PQR pass 
through the fixed points D, E, F respectively, and P lies on a fixed conic 
through E and F, whilst Q lies on a fixed conic through F and D, then B 
lies on a fixed conic through D and E. 

27. TA, TB are fixed tangents to a conic, and are cut by a variable 
tangent in the points m, m'. Shew that the locus of the circumcentre of the 
triangle Tmm' is a conic. 




EXAMPLES 155 

28. A, B are two fixed points on a conic, and the variable chords Am, 
Bm' intersect on a fixed straight line. Shew that the locus of the intersection 
of Am', Bm is a conic. 

29. If two triangles circumscribe a conic, their six vertices will lie on 
another conic. 

30. If two triangles are inscribed in a conic, their six sides will touch 
another conic. 

31. ABG is a given triangle, and PQR is a triangle of constant species 
inscribed in it. Shew that the sides of the latter triangle envelop three 
parabolas having the same focus. 

32. If a polygon of constant species moves in such a manner that three 
of its vertices move along three fixed straight lines which are not con- 
current, shew that the sides envelop parabolas all of which have the same 
focus. 

33. The base of a triangle touches a given conic, its extremities move 
on two fixed tangents to the conic, and the other two sides of the triangle 
pass through fixed points. Find the locus of the vertex. 

Let AB, AG he the two fixed tangents, D, E the fixed points, PQR any 
position of the triangle so that P is on AB, Q on AC, D on PR, E on QR. 
Since PQ is a tangent, (P) and (Q) are homographic ranges. .-. D (P) and 
E (Q) are homographic pencils, as are also D (R) and E {R). .: the locus of 
i? is a conic passing through I) and E. 



CHAPTER XII 

PASCAL — BRIANCHON — NEWTON — MACLAURIN 

146. Pascal's Theorem (1640). If a hexagon is inscribed in\ 
a conic the three points of intersection of the three pairs of opposite! 
sides are collinear*. 




Let ab'ca'bc' be a hexagon inscribed in a conic. The pairs of 
opposite sides are obtained by omitting one vertex in turn, and 
are (ab', a'b) ; (b'c, be) ; {ca, c'a). Let their intersections be 
respectively y, a, )8. These points shall be collinear. 

* For the history of this theorem, which was proved by Pappus for the 
line-pair, and was stated, but without proof, to be true for the circle by 
Pascal at the age of 15, see Art. 51. For a proof by the methods of ancient 
geometry, applicable both to the line-pair and conic, see Appendix II. 



146-148] pascal's theorem 157 

For by Art. 129 a {hh'c'a') = c (bb'c'a'). 

Therefore cutting these pencils by the transversals ba'j bc\ 

the range (bySa) = (bac'c). 

Therefore by Art. 23 the two ranges are in perspective, 
and consequently ya, 8c', and ea are concurrent, i.e. ya passes 
through f3. 

147. Conversely, if a hexagon has the three intersections of the 
three pairs of opposite sides collinear, it can be inscribed in a 
conic. 

For in Fig. 78, taking the hexagon to be ab'ca'bc' as before, 
the ranges (bySa) and (bac'e) have the point b common, and the 
lines ya, Sc', ac concurrent in jS. Hence the ranges are in per- 
spective, and are therefore equicross by Art. 21. 

.-. (6y8a') - (6ac'e), 

.'. a (bySa) — c (baCi). 

Therefore by Art. 138 the points a, b', c, a, b, d lie on a conic. 

148. We will give another proof of this important theorem. 
Taking a and a! as vertices, we have 

the pencil a (b' cbc') = a' (b' cbc) by Art. 129, 

= a(cb'cb) by Arts. 3 and 16. 

Therefore by Art. 59 the cross-centre of the two pencils 
a{b'cbc') and a' (cb'c'b) is on the line joining the intersection of 
the lines (ab', a'b') to the intersection of {ac, a'c), i.e. it is on 
the line b'c. 

Similarly it is on the line joining the intersection of {ab, a'b) 
to that of {ac, a'c), i.e. it is on the line be'. 

Therefore a, the intersection of b'c and be', is the cross-centre. 

Similarly it may be shewn to be on the line joining the inter- 
section of {ac', a'c) to that of {ab', a'b), i.e. a is on the line py. 



158 



CROSS-RATIO GEOMETRY 



[CH. XII 



Since a is the cross-centre of the pencils a (6'c6c') and a{ch'ch), 
it follows from Art. 58 that {aa, a a) are a pair of corresponding 
rays, as are also {a'a, aa). 

149. Brianchon's Theorem (1806). If a hexagon is cir- 
cumscribed about a conic, the three diagonals joining the three 
pairs of opposite vertices are concurrent. (The correlative of 
Pascal's Theorem.) 




Fig. 79. 



The hexagon is obey d' a. 

The range {abed) = {a'b'c'd') by Art. 1 30, 
- {b'a'd'c') by Art. 3. 

Therefore by Art. 50 the cross-axis passes through the inter- 
sections of {aa', bb') ; {bd' , ca') ; and {cc, dd'), i.e. the points 
a, p, y are collinear. 



J 



149-151] BRIANCHON — NEWTON — MACLAURIN 159 



Newton's Method of describing a conic (1687)*. 

150. Two angles aOP, aO'P, of given magnitudes a, /?, rotate 
about their vertices 0, 0' which are fixed. If the intersection a 
of two of their sides Oa, O'a moves along a fixed straight line L, 
the intersection P of the other two sides OP, O'P will describe 
a conic. 

o L 




Fig. 80. 

The pencils 0{a) and {P) are superposable, and therefore 
homographic. Similarly 0' {a) and 0' (P) are homographic. The 
pencils 0(a) and 0' {a), being in perspective, are, by Art. 45, 
homographic. Therefore by Art. 44 (P) and 0' (P) are homo- 
graphic, and consequently by Art. 138 the locus of P is a conic 
passing through and 0'. Of course, if the point a, instead of 
describing the straight line L, moves along a fixed conic passing 
through and 0', the pencils 0(a) and 0' (a) will be homo- 
graphic by Art. 129, and the locus of P will still be a conic 
through 0, O'f. 

151. Maclaurin's^ Theorem (1722). If the sides of a 
triangle aa'm pass through three fixed points P, Q, H, whilst 
two of tJie vertices a and a describe straight lines OL, 0L\ the 
locus of the third vertex m is a conic. 

* Principia, Bk i, Sect, v, Lemma 21. 

t Chasles, Apergu Historique, Note xv, Sect. 9 (1837). 

X Professor of Mathematics at Aberdeen 1717, and at Edinburgh 1725. 
" The one mathematician of the first rank trained in Great Britain in the 
18th century," Dictionary of National Biography. 



160 CROSS-RATIO GEOMETRY [CH. XII 

P 




Fig. 81. 

The ranges (a) and (a) are in perspective, centre F, and by- 
Art. 4:1 (1) are homographic. Therefore the pencils Q (a) and 
E (a'), i.e. Q (m) and R (m), are homographic. Hence by Art. 138 
the locus of m is a conic through Q and E. 

If PQ meets OL' in h\ and PE meets OL in c, the conic will 
evidently pass through the points 0, 6', c. 

152. If the points P, Q, E are in a straight line, let it meet 
OL, OL' in 6, h'. Then this line is a common ray of the two 
pencils, which are consequently in perspective, and the locus of m 
is a straight line. See also Chap. lY, Ex. 3. 

153. Maclaurin's Theorem can also be derived from Pascal's, 
for in Fig. 78 if we suppose a, b, c, a', c to be five fixed points, and 
b' movable, and consider the triangle yb'a, its sides pass through 
three fixed points, viz. a, fB, c, and two of its vertices move along 
the fixed lines ba', be'. 

We might also derive Pascal from Maclaurin, for in Fig. 81 
since by Art. 151 the locus of m is a conic through 0, Q, E, 
b', c, m, if we consider the inscribed hexagon Qb'OcEm, the inter- 
sections of pairs of opposite sides obtained by omitting one vertex 
in turn are (Qb', cE), i.e. P; (b'O, Em), i.e. a' ; (Oc, 7nQ), i.e. a; 
and these three points are in a straight line. 



152-155] CORRELATIVE OF MACLAURIN 161 

154. If the three vertices of a tHangle move on fixed straight 
lines, and two of its sides pass through fixed points, the third side 
will envelop a conic. (Correlative of Maclaurin.) 




If ama! is any position of the moving triangle, the ranges (a) 
and {a) are obviously homographic with (m), and therefore by 
Art. 39 with each .other. Consequently, by Art. 139, aa envelops 
a conic. 

By giving m different positions we see that the conic touches 
the five lines PS, QR, BS, OP, OQ, and it may be shewn that a 
similar relation exists between this proposition and Brianchon's 
Theorem as was shewn to hold between Maclaurin and Pascal, 
viz. that either of the two can be derived from the other. 

155. If two ranges of points on a conic, abc ..., ab'c ..., are 
Common such that the two conic-pencils formed by joining 

points of them to any other point on the curve are homo- 

conic-ranges, graphic, the ranges (i.e. the conic-pencils) abc... 
and ab'c ... are said to be homographic, and the points where 
the common rays of the pencils cut the curve are called the 
co7nmo7i points of the ranges. 

The student should carefully notice'^he difference between 
the homography of ranges on a conic, and "that of ranges on a 

M. H 



162 



CROSS-RATIO GEOMETRY 



[CH. XII 



straight line or lines. In the latter case the ranges are homo- 
graphic when their cross-ratios are equal, but we cannot speak 
of the cross-ratios of ranges of points on a conic. In the latter 
case we always imply the cross-ratios of the conic-pencils formed 
by joining the points of the ranges to some point or points on 
the conic. 




156. Given a range of points (abc ...) on a conic, to construct 
a range on the conic homographic to it. 

As in Art. 38 this can be done in an infinite number of 
ways ; for if we take ahc for the characteristic of the first range, 
we can take any three points a', 6', c' on the conic as the charac- 
teristic of the second range. 

If m is any variable point on the first range we can find its 
corresponding point as follows : 

As in Art. 146 construct the Pascal line ef of the hexagon 
ah'ca'hc'. This will pass through B and C, the intersections of 
{ab', ah) and {ac', a'c). Join a'm meeting ef in M. Then aM 
will meet the conic in the required point m'. 

For the pencils a {ahem) and a {ah' cm), i.e. a! {ABCM) and 
a {ABCM), are homographic by Art. 45. Therefore, by Art. 129, 
if V is any point on the conic, V{abcm) = V{a'h'cm'), i.e. the ranges 
{ahem,) and {a'h'c'in) are homographic. 



156-159] CONSTRUCTION OF HOMOGRAPHIC RANGES 163 
The Pascal line may be called the cross-axis of the two 



157. If we suppose the point m to coincide with e, the above 
construction shews that m! will also coincide with e, so that e is 
one of the common points of the ranges. Similarly y is the other 
common point. 

158. If the Pascal line ef is cut by the pencils V {ahem) and 
V{a'h'c'm') in the ranges (aySy/x,) and (a'yS'y'yu,'), these ranges are 
obviously homographic, and e, f are their common points. 

159. As in Arts. 75, 80, the cross- ratio of {aaef) is constant, 
where a, a is any pair of corresponding points on ef, and there- 
fore the cross-ratio of V {aaef) is constant, where a, a! is any 
pair of corresponding points and V any point on the curve. 

And conversely, if e, f are two fixed points on the co7iic, and 
a, a' two variable points on the curve such that the cross-ratio {aa'ef) 
is constant, a and a' will trace out two homographic divisions 
in which e,f are the common points, Cf. Arts. 192, 201 Cor. 
a b 




M' F MC B A E 

Fig. 84. 



11—2 



164 CROSS-RATIO GEOMETRY [CH. XII 

160. On a given straight line to construct a range homo- 
graphic to a given co-axial range. 

Let ABC be the characteristic of the given range. Take any 
three points A ', B', C on the given line to be the characteristic of 
the required range. Describe any conic, or circle, and on it take 
any point V. 

Join V to the different points of the characteristics, and 
produce the joining lines to meet the conic in a6c, a'h'c'. Con- 
struct the Pascal line ef of the hexagon ah'ca'hc. Let M be any 
point on the given range. It is required to find M\ the point on 
the given line corresponding to M. 

Join MV meeting the conic in m. Join a'm meeting the 
Pascal line ef in /x, and let a/x meet the conic in m. Then m V 
will meet the given line in the required point M'. 

For by Art. 156 the conic-pencils V {ahem) and V {a'h'c m') 
are homographic, and therefore so also are the ranges {ABCM) 
and {A'B'C'M') in which these pencils are cut by the given line, 
the common points being E, F where the given line cuts the rays 
Fe, Vf. See also Arts. 82—86. 



CHAPTER XIII 

POLE AND POLAR. CONJUGATE POINTS AND LINES. CIRCULAR 
POINTS AT INFINITY. DESARGUES' THEOREM AND ITS 
CORRELATIVE. PROPOSITIONS RESPECTING TRIANGLES, 
QUADRANGLES AND QUADRILATERALS INSCRIBED I>^ 
AND CIRCUMSCRIBED ABOUT A CONIC. CONTRA-POLAR 
CONICS 

161. If p is a fixed point in the plane of a conic, and any 
chord is drawn through it, the locus of the fourth harmonic oj 
p for the points in which the chord is cut hy the conic is a straight 
line*. 

Let paa' and pbh' be any two chords through p. Let aj), a'p' 
the tangents at a, a meet in P. Let ah, a'h' meet in m, and ah', 
ah in n, and let mn meet aa in a and hh' in /?. 

Then from the quadrilateral aa'h'h the ranges {paaa'), (ph(Bh') 
are in perspective, centre m, and (pa'aa), (ph^b') are in perspective, 
centre n, therefore as in Art. 118, a is the fourth harmonic of p 
for a, a' and similarly for ^. We will shew that the tangents at 
a, a' meet on mn f. 

The pencil a{ahh'a') = a' (ahh'a) by Art. 129, 

= a' (a'b'ha) by Arts. 3 and 1 6, 

and as the pencils have a common ray aa', they are in per- 

* Apollonius, Bk in, Prop. 37. 
+ Lahire, Bk ii, Props. 24, 27. 



166 



CROSS-RATIO GEOMETRY 



[CH. XIII 



spective, by Art. 45, and the intersections of (ah, a'h'\ {ah', a'b) 
and (aa, a'a'\ i.e. of the tangents ap, a'p, are collinear. Hence 
P lies on mn. 

Therefore for every chord hb' through p, the fourth harmonic 
)8 of p for the points, real or imaginary, in which the chord cuts 
the conic lies on the line Pa. 

We have proved incidentally that the line which is the locus 
of the fourth harmonics passes through 

(1) The cross-intersections such as {ah', ah). 

(2) The intersections of tangents at the extremities of the 
chords through p. 

(3) The points of contact of the tangents from p. These of 
course are imaginary if p is an internal point. 



I 




Fig. 85. 



161-162] 



POLE AND POLAR 



167 



Pole and Def. The point p and the locus of its fourth 

polar. harmonic are called pole and polar. 

It is obvious from the definition that every point has but one 
polar, and consequently every line has but one pole. 

162. If from any point P on a fixed straight line L we draw 
two tangents Pa^ Pa' to a conic, and a straight line L' the fourth 
harmonic of L for Pa and Pa, the line L' will always pass through 
a fixed point, viz. the pole of L. (The correlative of Art. 161*.) 

m 




Fig. 86. 
Let the chord of contact aa' meet Z in a and L' \n p. Then 
since P (aapa) is a harmonic pencil, (aapa') is a harmonic range. 
Therefore the polar of a passes through p. And the polar of P, 
i.e. aa, passes through p. Consequently p is the pole of L, and is 
therefore a fixed point. 

* Lahire, Bk ii, Props. 23, 26. 



168 CROSS-RATIO GEOMETRY [CH. XIII 

163. The intersection of two chords is the pole of the line 
joining their poles. 

In Fig. 85, let aa\ hh' be the chords. The pole P of the 
chord aaJ lies on mn, as does also the pole of hh' ; and the pole of 
mn is p, the intersection of the chords. 

Note, mn obviously divides both the chords aa', hh' har- 
monically for p. 

164. Any number of rays through a fixed point intersecting 
a conic determine on it two sets of divisions {i.e. a conic-pencil) in 
involution, whose double points are the points of contact of the 
tangents from the fixed point. 

In Fig. 85 if the polar of the fixed point p meets the conic in 
e, f pe, pf are the tangents from p, the polar of p is the axis of 
perspective of the pencils a' (ahc ...) and a(ah'c ...), and there- 
fore the divisions or conic-pencils [ahc ...) and {a'h'c ...) are 
homographic, and have e, f for their common points. They are 
also in involution, for if pVV is any chord through p, Fa, FV 
intersect on efhy Art. 161, as do also Va, V'a. Therefore ef is 
the axis of perspective of the pencils F (ahca') and V {a'h'c' a). 

Therefore by Art. 24 

V{ahca') =: V {a'h'c a) 

= V {a'h'c a) by Art. 129. 

Therefore any transversal is cut by these homographic pencils 
in two ranges which by Art. Ill are in involution, and conse- 
quently, the pencils V {ahc ...) and V {a'h'c' ...) are in involution, 
i.e. the divisions {ahc...) and {a'h'c',..) are in involution, e, f 
being the double points. 

Conversely if we have on a conic two sets of divisions forming 
a conic-pencil in involution, tJie lines joining corresponding points 
pass through a fixed point, and the polar of this point is the Pascal 
line of the system. 



163-166] CONJUGATE POINTS AND LINES 169 

Therefore, to construct a conic-pencil in involution, given two 
pairs of conjugate elements {aa\ hh'), in Fig. 85 produce aa\ hh' 
to meet in p. Then any chord through p will give a pair of 
conjugate elements, and the double points of the involution are 
the points where the polar of p cuts the conic. 



Conjugate points and lines. 

165. Def. Two points are said to be conjugate for a conic 
Conjugate when one lies on the polar of the other. 

PO"its. If P is a fixed point, the locus of its conjugate 

^ is a straight line, viz. the polar of P. 

Two conjugate points cannot both lie within the conic, for 
the polar of each would then be outside the curve and could not 
pass through a point lying within it. 

If P, Q are two external conjugate points, Q lies on the 
produced part of the chord of contact of tangents from P, and 
therefore PQ cannot cut the curve. 

Two lines are said to be conjugate for a conic when one passes 
Conjugate through the pole of the other. 

lilies. Any pair of conjugate lines through a fixed 

point form with the pair of tangents from the point a harmonic 
pencil. 

Of two conjugate lines, always one, sometimes both, meet the 
curve. 

A triangle is said to be self-conjugate for a conic when each 
Self-conju- vertex is the pole of the opposite side, 

gate triangle. It follows from the above that a self-conjugate 

triangle has one, and only one vertex within the curve, and the 
side opposite to it is entirely without the curve. 

166. It will be noticed that we have made frequent use of 
the word conjugate, viz. in the case of the two pairs of points in 
a harmonic range, in the case of two corresponding points in an 



170 CROSS-RATIO GEOMETRY [CH. XIII 

involution range, and again in the theory of pole and polar. 
A little consideration will shew the student that we are justified 
in doing so, and that the expression two conjugate points for a 
conic may be taken to imply that they possess the following 
properties : 

(a) The two points are such that each lies on the polar of 
the other. 

(/3) The two points are harmonic conjugates for two fixed 
points on the line joining them, viz. the points (real or imaginary) 
where the line cuts the conic, and 

(y) The two points are corresponding points in two homo- 
graphic co-axial ranges which together form a system in involution, 
the double points being the real or imaginary points in which the 
line joining the two conjugate points meets the curve. 

Similarly the expression two conjugate lines for a conic im- 
plies : 

(a) The two lines are such that each passes through the pole 
of the other. Hence a pair of conjugate diameters of a conic are 
conjugate lines, for each passes through the pole (at infinity) of 
the other. Also, any pencil of diameters is homographic to the 
pencil formed by their conjugates. 

(^') The two lines are harmonic conjugates for two fixed 
lines through their point of intersection, viz. the tangents (real 
or imaginary) from it to the conic. 

{y) The two straight lines are corresponding rays in two 
concentric homographic pencils which together form a system in 
involution, the double rays being the tangents (real or imaginary) 
from the common centre of the pencils to the conic. 

From the preceding articles it follows that : 

(A) The pole is the conjugate of every point on its polar, 
and the polar is the conjugate of every line through the pole. 

(B) One point, and one only (which may be at infinity), can 
always be found conjugate to each of two given points, viz. the 



166-169] CONJUGATE POINTS AND LINES 171 

intersection of their polars; and one line, and one only (which 
may be at infinity), can always be found conjugate to each of two 
given lines, viz. the line joining their poles. 

(C) The lines joining two conjugate points to the pole of the 
line drawn through them are the polars of the conjugate points, 
and are themselves conjugate lines ; and the points where two 
conjugate lines intersect the polar of their intersection are the 
poles of the conjugate lines, and are themselves conjugate points, 
i.e. in each case the assemblage of lines and points form a self- 
conjugate triangle. 

167. From Art. 166 (y) we obtain a very important property. 
In Fig. 85 since the ranges of points (P), (a) form a system in 

involution, they are homographic by Art. 96, and consequently so 
are the range (P) and the pencil p (a). Hence 

Given any number of 'poles on a straight line, tJie range which 
they form is hom,ographic with the pencil formed hy their polars for 
a conic. 

168. If we have given a pair of lines Z, L' which are not 
conjugate, and if any point P is taken on Z, it has one and only 
one conjugate point P' on L', viz. the point where the polar of P 
meets L' ; and if the points P, P' move along L and L\ they will 
form homographic ranges by Arts. 167 and 42. 

If we have given a pair of points P, P' which are not con- 
jugate, and if any line L is drawn through P, it has one, and 
only one, conjugate line L' passing through P', viz. the line 
joining P' to the pole of L ; and if the lines Z, L' rotate about 
P and P\ they will form homographic pencils, by Arts. 167 
and 42. 

169. We will now consider the two cases in which the line 
joining two conjugate points meets the conic in two (1) real, 
(2) imaginary points. 



172 CROSS- RATIO GEOMETRY [CH. XIII 




Fig. 87. 

(1) Suppose we have a conic intersected by a line L in the 
real points e, f, and let be the mid-point of ef. Then if P is 
any point on the line, and a another point on it such that ( Peaf) 
is a harmonic range, P and a are connected by the property that 
each lies on the polar of the other, and 

0P.0a=0e^=0f\ Art. 32. 

(2) If the line L does not meet the conic in real points, find 
C the centre of the conic, draw any chord pq parallel to L, and 
bisect pq by a diameter, meeting L in 0. If P is any point on Z, 
let the chord of contact of the two tangents from P meet L in a. 
Then P and a are conjugate points for the conic, and if E, F are 
the imaginary points in which L meets the conic, 

OE-^=OF'=^OP.Oa=OQ.O^^.... 

In each case P and a are conjugate points of an involution 
range, the first system being a non-overlapping one, and the 
second overlapping. In both the centre is real, and the value 
of the product OP . Oa is constant and real, being positive in the 
first and negative in the second. 



169-171] SELF-CONJUGATE TRIANGLE 



173 




170. 

conic. 



Fig. 88. 
To construct a triangle self-conjugate (Art. 165) for a 



Let be a point external to a conic. Its polar PQ meets 
the conic in two real points F, Q, and the other points on PQ 
are some of them internal and some of them external to the 
conic. Let p be any external point on PQ. The polar of p 
passes through 0. Let it meet PQ in p. Then Opp is a self- 
conjugate triangle, having each pair of its vertices conjugate 
points, and each pair of its sides conjugate lines. 

171. In Fig. 88 if PQ, P'Q' are a pair of conjugate li7ies 
which intersect within the conic, the conic-pencil [QPPQ') is 
harmonic. 

Let the chords PQ, PQ' intersect in p, and let the tangents at 
P, Q meet in 0. 



174 CROSS-RATIO GEOMETRY [CH. XIII 

Then {OP'pQ') is a harmonic range. 

Therefore Q {OP'pQ') is a harmonic pencil, and {QP'PQ') is a 
harmonic conic-pencil. 

172. If 0, p are a pair of conjugate points, OP, OQ, p'P, p'Q' 
the tangents from them, (QP'PQ') is a harm,onic conic-pencil, and 
conversely if {QP'PQ') is a harmonic conic-pencil, then PQ, P'Q' 
are conjugate lines, and their poles 0, p are conjugate points. 

This may be stated as follows : 

If two points are conjugate, their polars are conjugate lines ; 
and if two lines are conjugate, their poles are conjugate points. 

173. If PQ, P'Q' are a pair of conjugate chords, the tangents 
a,t their extremities cut any variable tangent in a harmonic range 
hy Arts. U\, 111. 

174. Given a fixed chord PQ, whose pole is 0, and a variable 
point r on the conic, rP, rQ will meet any line OP'Q' through in 
two conjugate points. 

For by Art. 166 PQ, P'Q' are conjugate lines, and by Art. 172 
the conic-pencil r {QP'PQ') is harmonic. 

Therefore the range {X'P'XQ') is harmonic, and consequently 
X, X' are conjugate points. 

Conversely, if X, X' are any pair of conjugate points on a 
line through 0, the lines PX, QX' intersect on the conic. In 
other words : 

If X, X' are a pair of conjugate points, P' , Q' the points 
where the line joining them, cuts the conic, and r any point on the 
curve, then if fX, rX' meet the conic in P, Q, the chords PQ, P'Q' 
are conjugate. 

175. PQ, P'Q' are a pair of conjugate chords meeting in p, 
and OP, OQ, p'P', p'Q' the tangents at their extremities are cut by 
a variable tangent in T, T', t, t'. Then (1) Ot, Of, (2) p'T, p'T', 
(3) pT, pT' are pairs of conjugate lines. 



J 



172-177] CONJUGATE POINTS AND LINES 175 

(1) By Art. 173 (^TtT't') is a harmonic range. 

Therefore 0{TtT't!) is a harmonic pencil, and consequently 
Ot^ Ot' are conjugate lines. 

(2) Since p (TtT't') is a harmonic pencil, p'l\ p'T' are con- 
jugate. 

(3) Let rP, rQ meet Op' in Z, Y'. 

Then Op is the polar of p (Art. 163), and rP is the polar of 
T, and therefore Y is the pole of pT. 

Similarly, since rQ is the polar of T\ Y' is the pole of p7". 
And since by Art. 174 7 and Y' are conjugate points, the polar 
of each passes through the other, i.e. pT passes through Y' which 
is the pole of pT', and pT' passes through F, which is the pole 
of p2\ Therefore pT, pT' are conjugate. Hence 

The pairs of lines joining 1\ T' to any point on PQ, external 
or internal, are conjugate. 

176. Ot, Ot' are a pair of conjugate lines, and OP, OQ the 
tangents from 0. A variable tangent at r meets these lines in t, t', 
T, T', and tP', t'Q' the second tangents from t, t' intersect in p. 
Then p lies on the polar of 0. 

For -l = 0{PtQt') 

= {TtT't') 

= pencil of polars r (PP'QQ'). 
Therefore the conic-pencil (PP'QQ') is harmonic, and the lines 
PQ, P'Q' are conjugate, i.e. p, the pole of P'Q', lies on PQ the 
polar of 0. 

177. The pairs of tangents to a conic from points on a given 
straight line determine an involution on any tangent to the conic. 

In Fig. 88 let Op be the given line, p its pole, r the point of 
contact of a variable tangent. Then if the tangents from any 
points Oi, Og, O3 ... on the given line meet the variable tangent 
at T„ T;- T„ T;; T„ T;, ..., {pT„ pT,'), {pT„ pT,') ... are pairs 



176 CROSS-RATIO GEOMETRY [CH. XIII 

of conjugate lines by Art. 175. Therefore p(T^T^, T^T;, ...) is 
an involution pencil, and {T/f^, T^T^, ...) an involution range. 

Otherwise, the chords of contact all pass through a fixed 
point, viz. the pole of the given line, therefore by Art. 164 they 
determine an involution range on the conic, &c. 

178. If two tangents are drawn to a conic, any variable 
tangent is divided harmonically by the two tangents, their chord of 
contact, and the curve. 

In Fig. 88 let OF, OQ be the given tangents, and let a variable 
tangent meet them in T, T', their chord of contact in R, and 
the curve at r. Let Or meet PQ in S. Then because the polar 
of passes through R, the polar of R passes through 0. But 
the polar of R also passes through r. Therefore the polar of A^ 
is Or. 

Therefore - 1 = the range (RPSQ) 

= the pencil (RPSQ) 
= the range {RTrT'). 

Since (RPSQ) is harmonic, it follows that 

Any chord of a conic is cut harmo7iically by any tangent and 
the line joining its point of contact to the pole of the chord. 



Circular points at infinity. 

179. The circular points have been defined in connection 
with orthogonal pencils in Art. 113. We will now shew how 
they are connected with the circle and the conic. 

By Art. 114 if a segment aa subtends a right angle at a 
point V, the pencil V (aa'ii') is harmonic. Hence, if V, V are 
any two points on the circle whose diameter is aa', 

V {aa'ii') = V {aa'ii'). 

Therefore by Art. 129 the six points a, a', i, i', V, V lie on a 



178-182] cmcuLAR points ' 177 

conic. And since this is true for all positions of F', the conic 
must be the circle on aa! as diameter. Hence 

Every circle passes through the points i, i'. 

This may also be shewn as follows. By Art. 166 (y) for any 
conic a pencil consisting of pairs of conjugate lines through any 
point forms an involution system whose double rays are the 
tangents through the point, their points of contact being on the 
polar of the point. Now let the conic be a circle, and let the 
point be its centre C. Then the pencil is orthogonal, the double 
rays are the lines Ci, Ci' which are consequently the asymptotes 
to the circle, the points of contact being % i' since the line at 
infinity is the polar of C, and as before we infer that all circles 
pass through these two points. 

If the circle is of indefinitely small radius, the above 
property leads us to infer that it coincides with its asymptotes 
Ci, Ci'. 

180. To obtain the converse, viz. that any conic through i, i 
is a circle, draw the tangents at i, ^'. Their intersection, having 
for its polar the line at infinity, is the centre, and the involution 
of conjugate diameters, having Ci, Ci' for double rays, is ortho- 
gonal, i.e. the conic is a circle. 

181. If the angle aVa' is not a right angle, describe any 
circle passing through V, and cutting Va, Va' in a, a'. Then 
since the circle passes through i, i', the pencil V (aa'ii') is constant 
for all points V on it. Also, if we suppose V to be fixed, and 
the chord aa' to vary in position whilst retaining its length, the 
condition that V {aa'ii') is constant is equivalent to the statement 
that aVa' is a constant angle. 

182. If the tangent at any point P of a conic meets the 
directrix corresponding to the focus S in the point Z, we know 
that the angle PSZ is a right angle, and SF, SZ are a pair of 
conjugate lines by Art. 166 (a'). Hence any pair of perpendicular 

M. 12 



178 CROSS-RATIO GEOMETRY [CH. XIII 

chords through a focus are conjugate lines, and a pencil formed 
of pairs of such chords, being orthogonal, is in involution, the 
double rays being by Art. 113 the lines joining S to i and i\ and 
by Art. 166 (y') these are the imaginary tangents from aS' to the 
conic, the points of contact lying on the polar of aS^, i.e. the 
directrix. This is sometimes expressed by saying " The focus 
is a point circle having double contact with the conic along the 
directrix." 

The converse of the above property is sometimes given as a 
definition, viz. Any point in the plane of a conic 
from which there can be drawn more than one pair 
of conjugate lines at right angles to one another is a focus. 

Since the tangents from i, i' intersect in a focus, there are in 
general four foci, two being real, and two imaginary. 

In the case of a parabola the circum-circle of a tangent- 
triangle passes through the focus, and since Si, Si' are tangents, 
the line joining i, i\ i.e. the line at infinity, is a tangent. 

183. If Ca, Cal are the asymptotes of a rectangular hyper- 
bola meeting the line at infinity in a, a', the pencil C (aaii') is 
harmonic by Art. 114, and therefore by Art. 166 {/3) i, i' are 
conjugate points for the conic. 

Conversely, any conic which has 2, % for conjugate points is a 
rectangular hyperbola. 

The triangle CO! is evidently self -con jugate for the rectangular 
hyperbola. 

184. Fregier's Theorem. If P is a fixed point on a conic, 
PQ, PR any pair of chords through P at right angles, the chord 
QR always passes through a fixed point on the normal at P. 

The pencil, centre P, is orthogonal, and therefore in involu- 
tion. Therefore by Art. 164 the chord QR passes through a 
fixed point. If the angle QPR is rotated about P until R coin- 
cides with P, QR becomes the normal at P. Hence the fixed 
point through which QR passes lies on the normal at P. 



183-185] CONJUGATE LINES THROUGH FIXED POINTS 179 

185. A and B are two fixed points, AP, BP lines through 
them conjugate for a given conic a. The locus of P is a conic p 
passing through A and B, and through the j^oints where their 
2)olars meet a. 




Fig. 89. 

Let OD, EF be the polars of A, B, and let c, c^ be a pair of 
conjugate points on CD. Let Ac, Bd meet in P. Then since 
BP passes throughj d, the pole of Ac, AP and BP are conjugate 
lines. And the range (c) = the range {d), therefore the pencil 
A (P) = the pencil B{P), and by Art. 138 the locus of P is a 
conic passing through A and B. The locus evidently passes 
through the points G, D E, F, and this indirectly gives us a 
proof of the property of Art. 199. 

Again, since P and d are conjugate points for a, the proposi- 
tion might be stated as follows : 

If through a fixed point B a transversal is drawn meeting a 
fixed line CD in d and a given conic a in Q, H, and on it is taken 
the point P siich that (PdQR) is harmonic, the locus of P is the 
conic /?. (See Art. 191.) 

Also, if from any point P on p tangents FT, FT' are drawn 
to a, since PA, PB are conjugate lines for a these form with FT, 

12—2 



180 CROSS-RATIO GEOMETRY [CH. XIII 

PT' a harmonic pencil, Art. 166 (j8'), which therefore cuts AB 
harmonically. Hence 

The locus of the intersection of tangents to a conic a which 
divide a given segment AB of a line harmonically is the conic (i. 

If the given line AB touches the given conic a, the locus /? 
degenerates into the line AB and the line joining the points of 
contact of tangents to a from A and B. 

If CD and EF meet in H, and if CD meets AB in G, then 
(x, ZTare conjugate points for a and /?, and H is the pole oi AB 
for both conies. 

Again, if on CD we take any point c, and on EF its con- 
jugate c' for a, this pair of points will trace out two homographic 
ranges, and the envelope of the line joining them is a conic yS' 
touching the lines CD, EF, and the tangents to a from A and B, 
This is the property of Art. 200 and may be stated as follows : 

If a chord of a given conic a is divided harmonically hy the 
conic and hy two given straight lines, its envelope is a conic fi^ 
touching the two given straight lines and the tangents to a drawn 
at the points where the two given lines intersect it. 

CoR. If in the first part of the proposition the points A, B 
are the circular points at infinity, since the tangents PT, PT' 
divide ii' harmonically, they are at right angles by Art. 113, and 
the locus p becomes the director circle. If in addition the conic 
a is a parabola, the locus degenerates into the line at infinity and 
the line joining the points of contact of tangents from i, i' , i.e. 
the directrix. 

Note. The student should insert the conic /? in Fig. 89, and 
notice that quadrilaterals can be circumscribed about a having 
the ends of two diagonals on y8, the third diagonal being fixed, 
containing the given segment AB; and the fixed point H, the 
pole of the third diagonal, is the intersection of the other two 
diagonals. Hence 



186] HOMOGRAPHIC DIVISIONS ON A CONIC 181 

If a, ^ are two conies such that quadrilaterals can be cir- 
cumscribed about a and inscribed in /?, then P is the locus of the 
intersection of tangents to a which divide harmonically the 
chord of /3 which lies on the third diagonal of any of the quadri- 
laterals. 

For these and other theorems of this chapter demonstrated 
analytically by means of the theory of invariants and covariants 
see Wolstenholme's Mathematical Problems, 3rd edition (1891), 
pp. 261—269. See also infra Chap. XIX, Exs. 22—24. 

186. If round two fixed points p, p' two straight lines rotate 
intersecting on a given conic at a, and cutting the conic again in 
a and a', the divisions (a) and (a) are homograj)hic, and their 
com/mon points are the points e, f where the line pp meets the 

conic*. 

a 




Fig. 90. 

In Fig. 90 take any point V on the conic. Then since aa 
passes through the fixed point p, the conic-pencils V (a) and V (a) 
are homographic by Art. 164. Similarly the pencils V (a') and 
V (a) are homographic. Therefore by Art. 44 the pencils V (a) 
and V (a) are homographic. If a is at f a and a will coincide 
at e, so that e is a common point of the divisions (a) and [a'). 
Similarly f is the other common point. 

* Chasles, Sections Coniqiies, Art. 229. 



182 



CROSS-RATIO GEOMETRY 



[CH. XIII 



187. Desargufs' Theorem (1593 — 1662). If a quadrangle 
is inscribed in a conic, any transversal meets its three pairs of 
opposite sides and the conic in four pairs of points in invo- 
lution. 

B, 




Fig. 91. 

A BCD is the inscribed quadrangle, 2)p' the transversal; 

By Art. 1 29 A {DBpp') = C {DBpp'), 
.'. {ahpp') = {h'a'pp') 

= ia'h'p'p) by Art. 3. 

Therefore by Art. 105, a, a' ; b, b' ; p, p' are in involution. 
Similarly by equating the pencils B^ADpp') and C {ADpp'), we 
find that b, b' ; c, c ; p, p are in involution. Hence by Art. 104, 
a, a' ; b, b' ; c, c ; p, p' form a system in involution. 

See also Art. 118 which shews that the first three pairs are 
in involution. 

The centre of the system can be found by the construction 
of Art. 102. We leave it to the student to shew that Desargues' 
Theorem can be obtained from Pascal's, and that each of them 
can be readily derived from the locus ad quatuor lineas, Art. 145, 
which is, as it were, the fons et origo of all these important 
properties. 

188. If a quadrilateral aba'b' circumscribes a conic, and if 
from any point P we draw pairs of lines to its opposite vertices, 
and also the tangents PQ, PR, these six lines will form a pencil in 
involution. (Correlative of Desargues' Theorem.) 



187-189] DESARGUES' THEOREM AND CORRELATIVE 183 




Fig. 92. 

Consider the tangents ah, ah', and let them be cut by any 
number of tangents mm\ nn' .... Then by Art. 1 30 the ranges 
{artm ... h) and (b'm'n ... a) are homographic, as are also the 
pencils F (amn ... 6) and F {b'm'n' . . . a), and their common rays 
are obviously the tangents FQ, FR. 

Hence F {ahQR) = F {h'a'QR) 

= F (a'h'RQ). 

Therefore, by Art. 105, F (aa', hb', QR) is a pencil in involution. 

Note. If ah meets ah' in c, and ah' meets a'h in c', the rays 
Fc, Fc' belong to the same involution. 

189. In Fig. 92 if the tangents a'h, a'h' move round the conic 
until the point a' coincides with a, the points h, h' will coincide 
with the points of contact £, B' of the tangents from a, and the 
theorem of Art. 188 becomes 



184 CROSS-RATIO GEOMETRY [CH. XIII 

If the sides of an angle BaB' touch a conic at B, B\ and if 
the sides of another angle QPB, touch the conic at Q^ R, then Pa 
is a double ray in each of the involution pencils P (QE, BB) and 
a (QP, BB'). 

190. If a quadrangle is inscribed in a conic, and a quadri- 
lateral circumscribed about it by drawing tangents at the 
vertices of the quadrangle, the two figures will possess the 
following properties : 

(1) Their internal diagonals will intersect in the same point G^ 
and form a harmonic pencil ; 

(2) Their third diagonals are in the same straight line, the 
polar of G, and their extremities form a harmonic range. 

(3) The three diagonals of the quadrilateral and the three 
diagonal points of the quadrangle form the same self conjugate 



(4) If aiiy transversal is drawn through any one of the three 
diagonal points E, F, G, the part intercepted either hy the conic 
or hy two opposite sides of either of the figures is divided harmonic- 
ally hy the diagonal point and its polar*, 

ABCD is the quadrangle, E, F, G its diagonal points. Then 
since by Art. 118 (1) (RAGC) is a harmonic range, FG is the 
fourth harmonic of FE for FA, FC. Hence, by Art. 161, Def., 
EF is the polar of G for the conic, and FG is the polar of E, 
and therefore the tangents at (A, C) and (.5, D) intersect on EF. 
Let these tangents form the quadrilateral prqs. 

Then p is the pole of ED, and G the pole of EF. Therefore 
pG i^ the polar of E, and is coUinear with GF. In the same way 
it may be shewn that q lies on GF. Similarly EG passes through 
r and s. Therefore the two figures have their internal diagonals 
passing through the same point G, and the intersections E, F, t, t' 

* Poncelet, Prop. Prqj. Vol. i, § ii, cap. ii, p. 97 Note, states that (1), (2) 
and (3) are due to Maclaurin, and (4) to Lahire. 



190] QUADRANGLE AND QUADRILATERAL 185 

t 



T E 




Fig. 93. 

of their opposite sides lying on the same straight line. Also 
these four points form a harmonic range, for in the quadrilateral 
prqs, by Art. 118 (1) the diagonals rs, pq divide the third diagonal 
tt' harmonically in £J, F. 

Since the range formed by the points t', t, F, E is harmonic, 
the pencil formed by their polars AG, BD, pq, rs is harmonic 
by Art. 167. 



186 



CROSS-RATIO GEOMETRY 



[CH. XII^H 
harmonio^^l 



Again by Art. 118 {EL AD) and (EMBC) are harmoni 
ranges. Therefore FG is the polar of E. Similarly EG is the 
polar of F. Consequently EF is the polar of G, and EFG is a 
self -conjugate triangle. 

(4) These properties follow at once by employing Desargues' 
Theorem and its correlative. 

Note. If from any point T on the third diagonal we draw 
tangents TP, TQ, then by Art. 162 T{PQGF) is -a harmonic 
pencil. Also by Art. 11 8 ( 1 ) (^ CGH) and {BDGK) are harmonic 
ranges. Therefore by Arts. 104, 111, T{AC, BD, PQ) is an 
involution pencil in which TG, TF are the double rays. 

191. Given a fixed point p, a fi^oced line L, and a conic C, 
let any transversal through p meet L in m, and G in a, a. Let 




Fig. 94. 



191-192] HOMOGRAPHIC DIVISIONS ON A CONIC 



187 



P be the pole of X, and fx the fourth harmonic of mfor a, a. As 
the transversal rotates about p, the locus of fx. is a conic, and the 
envelope of its polar is another conic*. (See also Art. 185.) 
Pfi is the polar of wi, therefore as m moves along L 
the pencil p (m) = the range of poles (m), by Art. 15 
= the pencil of polars P(/x), Art. 167. 
Therefore by Art. 138 the locus of /x is a conic through p, P. 
In Fig. 94 let mm' the conjugate of mp for C meet bb' the 
polar of p in m\ Then the pole of pm, being on mm\ is at m'. 
Also the range of poles (m') = pencil of polars p(ni)i Art. 167 

= the range (m), Art. 1 5. 
Therefore by Art. 139 mm' envelops a conic which touches L 
and the polar of p. 

192. Given two homographic divisions on a conic, the rays 
joining the pole of the Pascal line to the common points and to any 
pair of corresponding points form a pencil whose cross-ratio is 
constant. 




Fig. 95. 
Chasles, Sections Coniques, p. 136. 



188 



CROSS-RATIO GEOMETRY 



[CH. XIII 



Let ahc. . ., a'h'c' . . . be two homographic divisions. By Art. 1461 
draw the Pascal line meeting the conic in e,f. Then by Arts. 156, 
157 e,/aire the common points of the divisions. Let T be the 
pole of ef. Then by Art. 146, B the intersection of ab', ha' lies 
on ef, and the polar of B passes through jT, therefore T lies on 
the third diagonal of the quadrangle abh'a'. Therefore by Art. 
190 Note, T {ah\ ha', ef) is an involution pencil, the double rays 
being TB and the third diagonal through T. 

Therefore T {aa'ef) = T{h'hfe), by Art. 98 
= T{hh'ef), 

Conversely^ if ef is a given chord of a conic, T its pole, and 
a, a a pair of variable points on the conic such that T {aa'ef) is 
constant, {a) and {a') will mark out two homographic divisions of 
which e,f are the common points, Cf. Arts. 159 and 201 Cor. 

193. If two triangles ahc, def are inscribed in a conic C, their 
sides will touch another conic C*. 

d 




Fig. 96. 

By Art. 129, a{hcef) = d{bcef). 

Therefore the ranges which these pencils make on ef and he 
are equicross, i.e. (/3yef) = {hc€(f>). 

Therefore, by Art. 139, 6^, cy, ee, /</> are tangents to a conic 
touching Ohc and Oef 

* Brianchon, 1817. 



193-196] TRIANGLES AND CONICS 189 

194. If two triangles ahc, def are circumscribed about a conic 
C, their vertices will lie on another conic. 

In Fig. 96 by Art. 130 

{pyef) = {bc.<t>), 
.'. a (^yef) = d {bci<f>), 
.'. a (beef ) = d (beef). 
Therefore by Art. 138 the points b, c, e, f lie on a conic 
through a, d. 

195. If two conies C, C are such that 07ie triangle abc can 
be inscribed in C and circumscribed about C, then an infinite 
number of such triangles can be drawn. 

In Fig. 96 let b'c' be any chord of C which touches C. 
Through b\ c' draw the other tangents to G\ meeting in a'. Then 
by Art. 194 the point a lies on the conic through the five points 
a, b, c, b\ c', that is, it lies on C. 

196. Given a conic C and two self conjugate triangles abc, a'b'c'. 
Their six vertices lie on a conic C", and their six sides are tangents 
to a conic C". 



a 




Fig. 97. 



190 CROSS-RATIO GEOMETRY [CH. XIII 

Let dh' meet ac, he in a, /?, and let ah meet a'c', 6'c' in a', /8'. 
Then the pencil of polars c {aba'b') 

= range of poles (ba/S'd), by Art. 167 
= pencil c' (ba^'a) 
= pencil G {bah' a) 
= pencil c {aba'b'). 
Therefore, by Art. 138, a, 6, a', h' lie on a conic through c, c'J 
Again the range of poles {abaft') = pencil of polars c {bah' a) 

= pencil c {ft ah' a') 
— range {ftab'a') 
= range {afta'b'). 

Therefore, by Art. 1 39, aa, 5/3, a'a', b'ft' are tangents to a conic 
which touches ah and a'b'. 

197. If two conies G, C are such that one triangle abc which 
is self-conjugate for C can be inscribed in C, then an infinite 
number of triangles can be inscribed in C which are also self- 
conjugate for C. 




Fig. 98. 

Let a be any point on C and let the polar of a' for C meet C" 
in b', c. Then if a', b' are considered as two vertices of a triangh 



197-199] 



QUADRILATERAL AND CONIC 



191 



self -conjugate for (7, the third vertex will lie on 6'c'. But by 
Art. 196 the third vertex will lie on the conic through a, 6, c, a!^ h\ 
i.e. the conic C. Therefore the third vertex is at c. 

198. If two conies C, C are such that one triangle abc which 
is self-conjugate for C can be circumscribed about C\ then an 
infinite number of triangles can be circumscribed about C which 
are also self conjugate for C. 

Let b'c' be any tangent to C\ and let a be the pole of b'c for 
C, and let a tangent from a' to C meet b'c' in b' . Then if a', b' 
are considered as vertices of a triangle self -con jugate for (7, the 
third vertex c will lie on b'c. 

But by Art. 196 the third side a!c will touch the conic which 
touches a6, &c, c«, db\ b'c\ i.e. the conic C. 

199. If a quadrilateral PRQS circumscribes a conic, its 
points of contact and two of its opposite vertices lie on a conic. 




Fig. 99. 



P is the pole of ab, and c is the pole of QR. Therefore y is 
the pole of Pc. Similarly 8 is the pole of Pd. 



192 CROSS-RATIO GEOMETRY [CH. XIII 

The pencil of polars P (abed) = range of poles (abyS) 

= pencil Q (abyS) 
= pencil Q(abGd). 

Therefore by Art. 138 «, 6, c, (Z lie on a conic through P, Q. 

Similarly a, b, c, d lie on another conic which passes through 
i?, Sy and also on a third conic passing through the points 
(PJi, QS) and (PS, QE), not shewn in the figure. 

200. If a quadrangle abed is inscribed in a eonic, the tangents 
at its vertices and a pair of opposite sides are tangents to another 
conic. 

In Fig. 99 the range of poles (aby8) = pencil of polars P (abed) 

= pencil P (aped) 
— range (ajScd). 
Therefore aa, 6^, cy, dS are tangents to a conic touching ab, cd. 
There are two more such conies, viz. one touching be, ad, and 
one touching ac, bd. 

Note. The properties of Arts. 199, 200 are sometimes stated 
thus : 

If ab, cd are two chords of a conic, P, Q their poles, the six 
vertices of the triangles Pab, Qcd lie on one conic, and their six 
sides are tangents to another. 

201. If ab, cd are two chords of a conic, P, Q their poles, then 
if the conic-pencil (abed) = X, 

the pencil P (abed) = Q (abed) = X^. 
In Fig. 99 X = conic-pencil (abed) = c (abed) = (abyy), 
also X = d (abed) = d (aby'8) = (aby'S), 

.'. Xx X = (abyy) X (aby'S) 

= (abyS), by expansion, see p. 13, Ex. 3, 

= pencil of polars P (abed) = Q (abed), by Art. 199. 



200-202] 



CONTRA-POLAR CONICS 



193 



Cor. If {abc. . , ), {a'h'c . .,) are two homographic divisions on a 
conic, e, f their common points, (aa'ef) is constant, and conversely, 
if e, f are two given points on a conic, and a, a' a pair of variable 
points on the curve such that (aa'ef) is constant, (a) and {a') will 
mark out two homographic divisions of which e, / are the common 
points, by Art. 192. See also Art. 159. 

Contra-polar conies. 

202. Two conies a, /3 intersect in A, B, I, I', and the poles 
of one of the chords as II' for a and jS are T, C respectively. 
Then if TA is a tangent to jS, 

(1) TB will be a tangent to (3, 

(2) CA and CB will be tangents to a. 




Fig. 100. 



13 



194 CROSS-RATIO GEOMETRY [CH. XIII 

(1) Let AB, II' meet in D. Join CT meeting J 5 in E and 
//' in F. Then by Art. 163 the polar of D for a must pass 
through T and cut II' and AB harmonically. Similarly the 
polar of D for /S must pass through C and divide the same two 
chords harmonically. Hence CT is the polar of D for both 
conies. And AT is the polar of A for p. Therefore T is the 
pole of AD for /3, i.e. TB is the tangent to (3 at B. 

(2) For /?, C is the pole of //', and A is the pole of AT, 
therefore M is the pole of CA. Hence by Art. 161 {II'MN) is 
harmonic, therefore for a, M and N are conjugate points, and T 
being the pole of MN, TM is the polar of iV, i.e. NA touches a at 
A. And AN passes through G. Therefore GA is the tangent 
at A. Similarly it may be shewn that GB is the tangent at B. 

203. Def. Since the poles of a pair of common chords of a, j8 
Contra-polar for one of the conies are also their poles for the other 
conies. Poles, conic when the chords are taken in the contrary 
order, we shall call conies which are so related contra-polar conies, 
and the points (7, T their poles. 

From Art. 202 we obtain a simple method of describing a 
conic contra-polar to a given conic a. Take any point G outside 
a, draw a tangent GA, and join G to any two points /, /' on the 
curve. Then the conic which passes through A and touches 
Cly CI' at the points /, /' is the conic required. 

204. In two contra-polar conies the tangents at any one of 
the points of intersection, as A, divide the opposite chord II' 
harmonically. 

Let the tangents at A meet //' in i/, N, and let TA meet 
a again in A'. Then for a, TA, II' are conjugate lines, and by 
Art. 171 the conic-pencil {II' A A') is harmonic. 

Therefore A {II' AA') is harmonic, and therefore also {II' NM), 
i.e. the tangents at A divide //' harmonically. 

Similarly it may be shewn that //' is divided harmonically by 
the tangents at B, and AB by the tangents at / and /'. 



203-205] CONTRA-POLAR CONICS 195 

From the property of this article these conies might be called 
harmotomic (on the analogy of orthotomic), a term which would 
help to remind the student of the connection between them and 
orthogonal circles, see Chap. XIX, Data 23, and Examples 
51—61. 

205. In two contra-polar conies, a, /?, any transversal through 
either of the poles is divided harmonically by the two conies. 

Let a transversal through 2' meet a in F, Q. We will shew 
that F, Q are conjugate points for fi. 

Join CQ meeting a again in i?, and //' in K. Join RF 
meeting //' in L and AB in L'. 

Then FQ and //' are conjugate lines for a. 

.*. the pencil R{QFII') is harmonic, i.e. {KLII') is harmonic. 
.*. K, L are conjugate points for y8, 
i.e. X is the pole of (7^ for y8 (1). 

Again QR and AB are conjugate lines for a. 
.*. the pencil F(QRAB) is harmonic, i.e. {K'L'AB) is harmonic. 
.*. K\ L' are conjugate points for y8, 
i.e. Z' is the pole of r^' for y8 (2). 

.-. from (1) and (2), by Art. 163, Q is the pole of LL' for ft 
and therefore the polar of Q passes through F. Hence F and Q 
are conjugate points for /?, as we had to prove. 

CoR. If //' is a given chord of a conic ft C its pole, and if 
any transversal is drawn through C, and on it are taken a pair of 
conjugate points Q, R, then any conic through the four points 
Q, R, I, I' is contra-polar to the given conic jS. Consequently, if 
we have a system of conies through four given points /, /', Q, R, 
a conic which passes through two of them, as /, /', and is contra- 
polar to one of the conies, is contra-polar to every conic of the 
system, and its pole for //' lies on QR, the corresponding common 
chord of the system. 

13—2 



196 CROSS-RATIO GEOMETRY [CH. XIII 

206. If we have given four points a, b, c, d on a. conic a, 
and if the tangents at these points form a quadrilateral whose 
opposite vertices are P, Q ; i^, S ; T, U, it is clear that through 
the four points a, b, c, d three conies can be described contra-polar 
to a, viz. those having for their poles P, Q ; R^ S ; and T^ JJ. 

207. If on a conic a we take four points forming a harmonic 
c. p. as {A A' II') in Fig. 100, the poles being N, T^ the contra- 
polar conic ^ through the four points degenerates into the lines 
AA\ II, the tangents to (i at /, /' passing through N, and those 
at ^, A' passing through T. 

Note. Prof. A. Lodge has pointed out to me that the relations 
between contra-polar conies can be readily established by means 
of Art. 201. 

(1) If X = c.p.{ABir)iova,X^ = C{ABir), 
and if fx = c. p. {ABU') for jB, ix^ = G (ABIF). 

.'. A.2 = /A^. Now since A 4=//,, X must = — /a, and the condition 
that two conies through four given points should be contra-polar 
is X + /A = 0. 

(2) To prove Art. 204 we have 

X = (ABIF) for a = A {ABII') = (MDII'). 

.-. ^=(DMir) by Art. 3. 

-X = fji = (ABIF) iov P = A {ABIF) = {NDIF). 

.-. - 1 = - A X 1 - {NDIF) X {DMIF) 

A 

= {NMII') by expansion. 

(3) In the particular case when A, = — 1, and the conic-pencil 
for a is harmonic, ii = +\, and P degenerates into the two lines 
AB, IF. 

For since C {ABII') = 1, one pair of rays CI and CI' coincide 
by Art. 8. Hence the tangents to P at /, /' which have to 
pass through C coincide with the line //'. 



206-208] 



CONJUGATE POINTS AND LINES 



197 



Similarly since T{ABII')=l, TA, TB coincide, and the 
tangents to /3 at A, B must lie along ^^, i.e. ^ is the two lines 
AB, IF. Art. 207. 

208. The following useful properties in connection with con- 
jugate points and lines are given here for convenience of reference. 

Other properties will be found in Exs. 1 — 7, 35, 36, in which 
the letters have been so arranged that Fig. 101 will apply to 
them. 




AB^ II' are two chords of a conic intersecting in H ; C, T 
their poles. CT meets AB, II' in G, F, and the curve in iV", 0. 
The tangents at ^, ^ meet //' in K, L. 



198 CROSS-RATIO GEOMETRY [CH. XIII 

1. If TPQ is any chord through T, 
(a) PQ, II' are conjugate lines. Art. 166 (a'). 
{(i) the c. p. (PQir) is harmonic. Art. 171. 

If B is any point on the conic and i?P, BQ meet //' in D, 

(y) the range (II'DE) is harmonic, and consequently 
D, E are conjugate points, TD, TE are conjugate 
lines and TDE is a self-conjugate triangle. Art. 
174. 

(8) Also, if D, E are two conjugate points on //', and 
PR is any chord through one of them, Z>, 
T {II' , PB, DE) is an involution pencil, in which 
TD, TE are the double rays. Art. 118. 

2. (e) The ranges {ABGH) and (IPFH) are harmonic. 

Art. 163. 

(^) T{AB, II', KL, CH) is an involution pencil, in which 
TC, TH are the double rays. 2 (c). 

[r]) If TA meets //' in J/, K and M are conjugate points. 
Art. 171. 

3. {&) If P is a variable point on the curve, the double rays 

of the involution pencil P {II\ AB) always pass 
through N and 0. Art. 166 (a). 

EXAMPLES. 

[In Exs. 1—7, 36, 36 the letters are the same as in Fig. 101.] 

1. ir is a given chord of a conic, T its pole. J^ is a point on /I', 
and KA, KA' are two tangents cut by the variable tangent at B in the points 
C, C Then T {II' GC) is harmonic. (Chap. XIX, Ex. 10.) 

2. TPQ is a chord through T. The tangent at any point A meets 11' 
in K. M is the harmonic conjugate of K for I, I'. PM meets the curve in 
P'. Then PA is a double line of the involution pencil P (II 'QP') and QP' 
passes through K. (Chap. XIX, Ex. 15.) 



EXAMPLES 199 

3. Given a chord W, T its pole, and any chord PR cutting IV in D. 
Divide PR at S so that (PRSD) is harmonic, and let TS meet W in E. 
Then {II'DE) is harmonic. Conversely, if (II'DE) is harmonic, so also 
is (PRSD). 

Also, TD, TE are the double rays of the involution pencil T {II ', PR, BE). 
(Chap. XIX, Ex. 6.) 

4. ir is a chord of a conic, T its pole, D a fixed point on 11', If DPR, 
any chord through D, is divided harmonically at D, S, the locus of S 
is a straight line through T; and if this line meets 11' in E, the range 
{II'DE) is harmonic. 

5. II', AB are two given chords of a conic, T, C their poles. AB meets 
II' in H, and {II'FH) is harmonic. TPQ is a chord through T. Then II' 
is one of the double rays of the involution pencil F(AB, PQ). 

6. D, E are two conjugate points, R any point on the conic. If RD, 
BE meet the curve in P, Q, then PQ and DE are conjugate lines. 

7. 11' is a given chord of a conic, T its pole, TA'A a chord meeting 
ir in ill. The tangent at any point B meets II' in L, and V is taken on 
II' so that {II'LV) is harmonic. ^F meets the curve in W. Then AB is a 
double line of the involution pencil A {II', A'W) and A'W passes through L, 

8. ir is a chord, T its pole, TAB, TA'B' two chords through T. If 
^^', 5P' meet in G, and ^J5', A'B meet in D, the points G, D will lie on II', 
and divide it harmonically. (Chap. XIX, Ex. 20.) 

9. AB is a chord, G its pole. Through B a straight line is drawn 
meeting the conic in D and AG in E. The tangent at D meets AG in F. 
Then {AEGF) is harmonic. 

10. Given three tangents to a conic, draw a fourth tangent so that the 
part intercepted between two of the given tangents shall be bisected by 
the third. 

[Let or, OT' be the two given tangents, P the point of contact of the 
third. Join OP and produce it to meet the conic again in P' . The required 
tangent is parallel to the tangent at P'.] 

11. AB is a chord of a conic, G its pole, T the focus. Then AB is 
divided harmonically by GT and the directrix. 

12. Given a chord AB and its pole G, if the internal bisector of the 
angle AGB meets AB in D, and EF is any chord through D, then GD bisects 
the angle EGF. 

13. In Ex. 12 the poles of all chords through D lie on the line through 
G perpendicular to GD, 



200 CROSS-KATIO GEOMETRY [CH. XIII 

14. P, Q are two conjugate points. P moves on a fixed straight line 
whose pole is E, and PQ passes through a fixed point S. Shew that the 
locus of Q is a conic through B, S. 

15. P, Q are two conjugate points. P moves on a fixed straight line 
whose pole is R, and PQ subtends a constant angle at a fixed point T. 
Shew that the locus of Q is a conic through R, T. 

16. PQ, P'Q' are two conjugate chords, PR, any chord through P, 
meets P'Q in A, QQ' in B and Q'P' in G. Shew that (RABC) is harmonic. 

17. 11' is a chord of a conic, T its pole, A a fixed point in the plane of 
the curve. Any chord BG through A meets 11' in D, and (BGDP) is harmonic. 
Shew that the locus of P is a conic through A and T. 

[For let TP meet II' in E, Then the polar of D passes through P and 
T, and therefore also through E. And D, E, being conjugate points in an 
involution range, form two homographic divisions. 

Hence A{D) = T (E), i.e. A{P) = T (P). .-. &c. , Art. 138.] 

18. If two chords are drawn through any point, the lines joining their 
extremities meet on the polar of the point and divide it harmonically. 

19. In a hyperbola the portion of any tangent intercepted between the 
asymptotes is bisected at the point of contact. 

^20. Any tangent to a conic cuts the six sides of an inscribed quadrangle 
in an involution range of which the point of contact is a double point. 

21. TP, TQ are two tangents to one branch of a hyperbola. They and 
the chord of contact PQ intersect an asymptote in H, K, L respectively. 
TH, TK, TL cut the other asymptote in W, K', L', and a parallel through 
T to the first asymptote cuts the second in X'. Prove that {H'L'K'X') is a 
harmonic range, and that H'K, PQ, HK' are parallel. 

22. The straight line PP' is the normal chord at P. The chords PQ, 
PQ' are equally inclined to PP'. Shew that P'Q and P'Q' are harmonic 
conjugates to PP' and the tangent at P'. 

23. If through a fixed point on a conic two chords are drawn making 
equal angles with a fixed line, the chord joining their extremities will pass 
through a fixed point. 

24. If a triangle is self-conjugate to a rectangular hyperbola, its circum- 
circle passes through the centre of the conic. 

[In Art. 196 take GW for one of the triangles.] 

25. If a triangle is inscribed in a rectangular hyperbola, its nine-point 
circle passes through the centre of the conic. 

[Shew that the pedal triangle is self-conjugate to the conic] 



EXAMPLES 201 

26. Shew that two conies which have the same focus and directrix may 
be considered as having double contact. 

27. The circumcircle of a tangent triangle of a parabola passes through 
*he focus. 

[In Art. 194 note that SW is a tangent triangle.] 

28. is a fixed point. A variable chord through meets a conic in 
a, a', and on the chord is taken a point P such that [OPaa') is constant. 
Shew that the locus of P is a conic. 

29. The locus of the intersection of two tangents to a given conic drawn 
from corresponding points of an involution range is a conic passing through 
the double points of the involution and through the points of contact of the 
tangents to the given conic drawn from the double points. See Art. 185. 

30. The envelope of a chord of a given conic whose extremities lie on 
corresponding rays of a pencil in involution is a conic touching the double 
rays of the involution pencil, and also touching the tangents drawn to the 
given conic at the points where the double rays meet it. See Art. 185. 

31. Two fixed tangents to a conic meet a fixed line in ^, jB. A variable 
tangent meets the fixed tangents in Q, B, and the fixed line in S, and on it 
is taken a point P such that (PQRS) is constant. Shew that the locus of P 
is a conic passing through A and B. 

32. A tangent to a conic cuts two conjugate lines OP, OQ in P, Q. 
Shew that the other tangents from P and Q intersect on the polar of 0. 

33. ^ is a fixed point on a conic, and BCD an inscribed triangle. Any 
transversal through p cuts PC, CD, DB in a', h' , c', and the conic in p'. 
Shew that {p'a'h'c') is constant. 

[In Fig. 91, since {pp', aa', hh', cc') is an involution range, 
{jp'a'h'c') = {pahc)=A {pahc) = A (2?-DPC') = const.] 

34. If the extremities of two diagonals of a quadrilateral are conjugate 
points for a conic, the extremities of the third diagonal will also be conjugate; 
and if two of the three pairs of opposite sides of a quadrangle are conjugate 
lines, the third pair will also be conjugate. Hesse (1840). See Chasles, 
Sect. Con., Arts. 133, 134. 

35. ir, AB are two chords of a conic, T, C their poles. TPQ is any 
chord through T, and AP, BQ intersect in X. Shew that the locus of X is 
a conic passing through A and B and contra-polar to the given conic. 

[In Fig. 101 join TB meeting the conic in Y. 

Then since T is a fixed point, P, Q are conjugate points in an involution 



202 



CROSS-RATIO GEOMETRY 



[CH. XIII 



range. .". {P) = {Q). .'. A{P) = B {Q). .: the locus of Z is a conic through 
A and B. It also evidently passes through 7, I'. 

Now when P is at B, Q is at Y. :. the ray corresponding to AB ia BY, 
i.e. the tangent at B passes through T. .: by Art. 202, the conies are 
contra-polar.] 

36. II' is a chord of a conic, T its pole. TPQ is any chord through T. 
PU, QV &re two chords intersecting in X, and Y is the pole of UV. Shew 
that the conic through the five points U, V, X, I, I' is contra-polar to the 
given conic. 

37. In Fig. 100 if CT meets the conic a in u, v, shew that Au, Av meet 
j8 in two points u', v' which are such that the line joining them passes 
through G and D. 

38. In Fig. 100 if any transversal through G meets AB in W, shew that 
any conic through the points C, W, I, I' will be contra-polar to the conic a. 

39. In Fig. 100 if a system of conies passes through the four points 
Q, B, I, I', and from G pairs of tangents are drawn to all the conies, 
the points of contact will all lie on the conic ^. 



CHAPTER XIV 

COMMON CHORDS AND COMMON TANGENTS OF TWO OR 
MORE CONICS (1) PASSING THROUGH FOUR POINTS, 
(2) TOUCHING FOUR STRAIGHT LINES, i.e. OF PENCILS 
AND RANGES OF CONICS 

Two conies circumscribing a quadrangle. 

209. Two conies will in general intersect in four points 
which form an inscribed quadrangle. 

Defs. The line joining any two of the points of intersection 
Pair of ^^ ^^^ conies is called a common chord of the conies, 

common Of the six lines which can be so drawn, any 

chords. ^^Q whose point of intersection does not lie on the 

conies are called a pair of common chords. 

When we use the word pair in this restricted sense we shall 
print it in different type. It is obvious from Fig. 102 that there 
are in general three pairs of common chords. 

The point of intersection of a pair of common 
Chord vertex. 

chords we shall call a chord vertex*. 

In what follows, the three chord vertices will always be 
denoted hy P^, F2, Ps. 

* Suggested by Prof. A. Lodge. 



204 



CROSS-KATIO GEOMETRY 




Fig. 102. 

210. The distinctive property of a common chord of two 
conies is : 

The two polars of any point on a common chord intersect on the 
chord. 

For if P is any point on g^g^ a common chord of two conies A 
and jB, and if P' is the fourth harmonic of P for ^i and g^^ the 
polars of P for both A and B will pass through the point P' by 
Art. 161. 

211. The intersection of a pair of common chords has the same 
polar for both conies. 

For if in Fig. 102 g^g2 and g^g^ are the common chords inter- 
secting in Pi, the two polars of Pj considered as a point on g^g^ 
intersect on g^g^ by Art. 210. Considered as a point on g^g^, 
they intersect on this chord also. Hence they must coincide. 
Consequently, 



I 



210-215] COMMON CHORDS 205 

If P\i P25 Pa ^^<3 the three chord vertices, P^P^P^ is a self- 
conjugate triangle common to both conies. 

212. Any point Q which has the same polar for both conies is 
a chord vertex. 

For join Q to g, one of the points where the conies intersect, 
and produce Qg to meet the conies again in g'j g", and the common 
polar in Q'. Then by Art. 161, Def., {QQ'gg') and (QQ'gg") are 
harmonic ranges. 

Therefore g' and g" coincide. Hence Q lies on one common 
chord. Similarly it lies on the other common chord of the pair, 
and therefore coincides with a chord vertex. Consequently 

Two conies can have only one common self-conjugate triangle. 

213. The five pairs of points in which any transversal meets 
the two conies and the three pairs of opposite sides of the common 
inscribed quadrangle form, a system in involution. 

By Desargues' Theorem, Art. 187. 

CoR. If the transversal is a comm^on tangent, the points of 
contact are harmonic conjugates of the points where it cuts the 
opposite sides of the quadrangle, being the double points of the 
involution. 

214. In Fig. 102 i/" the polar of P^ meets the conies A and B 
and the pair of common chords through P^ in aa', bb', pp', these 
pairs of points will form a system in involution whose double 
points are P^ and P^. 

By Desargues. 

215. If three conies have one chord common to all, their three 
other corresponding common chords are concurrent. 

Let ^, ^, C be the conies, and let G be the chord common 
to all, and let G-^ be the corresponding common chord for A and 
B, G^ for A and C, and G^ for B and C. Let G^ and G^ intersect 



200 CROSS-RATIO GEOMETRY [CH. XI\' 

in g^. Through g-^ draw any transversal meeting the conies in 
a, a ] b, V ; c, c', and meeting G and G^ in g, g^. Then it is 
evident that g^ coincides with g^, for by Art. 213 (aa', bb', cc) is 
an involution range, in which (g, g^ and (g, g^) are pairs of con- 
jugate points. In other words: Given a system of conies through 
four fixed points, the corresponding common chord of any conic 
of the system with a fixed conic through two of the given points 
will pass through a fixed point. 



Pencil of conies. 



Pencil of Def. A system of conies circumscribing a 

conies. quadrangle is called by Chasles a pencil of conies*. 

216. Given a pencil of conies, any transversal will intersect 
them in a system of pairs of points in involution, and the double 
points of the system will be conjugate points for each of the conicsf. 
See Art. 213. 

Since the double points are the harmonic conjugates of each 
pair of points of intersection, the polar of one passes through the 
other by Art. 161, Def. 

By Arts. 211, 212 a pencil of conies has one, and only one, 
common self-conjugate triangle. 

Also, since Pj is the pole of P^Pzi the poles of chords through 
Pi lie on P2P3. 

217. A given pair of points are conjugate for only one conic 
of a pencil. 

For if P, P' are the points, the transversal PP' cuts the pencil 
in a range in involution of which the double points Q, Q' will 
not in general coincide with P, P'. By taking on PP' pairs of 

* Sect. Con., Art. 306. 

t This proposition was enunciated for a system of three conies by 
Sturm, 1826. 



I 



216-218] PENCIL OF CONICS 207 

points harmonic conjugates for P, P\ we can obtain a range in 
involution having P, P' for double points. By Art. 35 we can 
find on the transversal one and only one pair of points a, a' 
which will divide harmonically both the segments PP\ and QQ\ 
and which will therefore be conjugate points in both involutions. 
Then the conic of the pencil which passes through a will also 
pass through a', and will have P, P' for conjugate points. 

218. In a pencil of conies the polars of any point P will all 
pass through the same point P'*. 

Let the polars of P for two conies A, B oi the pencil intersect 
in F, and let PP' meet A in a, a' and B in h, b'. Then P and P' 
are the double points of the involution range of which the 
characteristic is (aa', hh'). Let a third conic C of the pencil cut 
the transversal PP' in c and c'. Then by Desargues' Theorem, 
Art. 187, c, c' belong to the involution {aa\ hh'). Therefore the 
range {cc'PP') is harmonic, and by Art. 161 the polar of P for C 
passes through P', Hence 

If two points are conjugate for two conies A, B, they are con- 
jugate for the pencil of conies to which A and B belong. 

If P, P' are a pair of conjugate points for a pencil, it is 
evident that PP' is a common tangent of the two conies of the 
pencil which pass through the points P, P' respectively. 

From this it follows that a system of conies passing through 
three given points ^i, g^, g^ and having a given pair of conjugate 
points P, P', pass through a fourth fixed point g^, viz. the fourth 
point of intersection of the two conies round g^g^g^ which touch 
the line PP' at P and P' respectively, and therefore the system 
constitutes a pencil. In other words : 

If three points are given on a conic, a pair of conjugate points 
are equivalent to a fourth point on the curve. 

* Lam^, 1818. 



208 CROSS-RATIO GEOMETRY [CH. Xr 

It can also be shewn that 

Two points on a conic and two pairs of conjugate points 
are equivalent to four points on the curve, i.e. determine a pencil 
of conies. 

For let the lines PP' and QQ' meet in A, and on them take 
the points A-y, A^ such that {PP'AA^ and (J^Q'AA^ are harmonic. 
Then the conic through g^g^AA^Ac^ obviously belongs to the 
system. Again, let the line g^g^ meet PP' in JB and QQ' in C. 
On PP' take B' such that [PPBB') is harmonic, and on QQ' take 
C such that (QQ'CC) is harmonic. Then the line-pair g-^g^ and 
B'C passes through the points gi, g^, B, B', G, C and therefore 
is a conic belonging to the system. Let these two conies inter- 
sect again in ^3, g^. Then since P, P' are conjugate for both 
conies, they are conjugate for the pencil to which they belong, 
and similarly for Q, Q' . Hence the system forms a pencil 
through the points gi, g^, g^, 9^- 

Two points and three pairs of conjugates determine a conic. 

For if we take a third pair of points R, B' , by Art. 217 only 
one conic of the pencil will have B, B' for a pair of conjugates, 
unless R, B' are themselves conjugate points of the system. 

219. If a point P move along a fixed line L, its conjugate P 
for a pencil of conies will describe a conic passing through the 
poles of L for the pencil*. 

By Art. 219 it is only necessary to consider two of the conies, 
A and B. 

Let Z^, Iji be the poles of L for A and B. Then P'l^ and P'l^ 
are the polars of P for A and B. And by Art. 167, as P moves 
along X, 

the range of poles (P) = the pencil of polars Ij^ {P') 
and also — the pencil of polars l^ (P'). 
Therefore by Art. 44, 

the pencil Ij^ (P') = the pencil l^ {P'). 
* Poncelet, Prop. Proj. Vol. i, Art. 370. 



219-221] ELEVEN-POINT CONIC 209 

Therefore by Art. 138 the locus of P' is a conic passing 
through ^4 and l^. Now A and B are any two conies of the 
pencil. Therefore the locus of P' passes through the poles of L 
for all the conies of the pencil. 

„ , , Def. The point P' in which the polars of P 

Reciprocal . ^ , ^ 

point. intersect is called the reciprocal of the point P, 

Reciprocal and the locus of P' is called the reciprocal conic 

of L for the pencil*. 

220. The cross-ratio of the pencil formed by the polars of any 
point P for four conies of a pencil of conies is independent of the 
position of P. 

Let P' be the reciprocal of P. Let Q be any other point, Q' 
its reciprocal. Let L be the line PQ, l^, Ib-" its poles. 
Then by Art. 219 

i.e. the cross-ratio of the pencil formed by the polars of P is equal 
to that of the pencil formed by the polars of any other point. 

Note. This cross-ratio is called the cross-ratio of the four 
conies circumscribing the given quadrangle f. 

CoR. The cross-ratio of the tangents at a point of inter- 
section of four conies of a pencil is the same as that of the 
tangents at each of the other common points. 

221. In Art. 219 if L meets the line P2P3 in Q, the polars 
of Q will pass through P^. Hence P^ is a point on the reciprocal 
conic, and similarly for the points P^ and P3. Consequently 

The reciprocal conic of any line for the pencil passes through 
the vertices of the common self conjugate triangle. 

* Poncelet, Prop. Proj. Vol. i, Arts. 82, 370. 
t Chasles, Traite des Sect. Con., Art. 325. 

M. 14 



210 CROSS-RATIO GEOMETRY [CH. XIV 

The reciprocal conic of L for the pencil passes through the 
following eleven points : 

(1) The three vertices of the common self-conjugate triangle. 

(2) The six points which are the harmonic conjugates of the 
points where L cuts the sides of the quadrangle. 

(3) The double points of the involution on L determined by 
the pencil. 

For the above reasons the reciprocal conic is sometimes called 
the eleven-point conic of L. 

CoR. If L passes through one of the vertices of the common 
self -conjugate triangle, as Pj, the reciprocal conic degenerates 
into two straight lines*, viz. 

(1) The line P^Pz, which contains the poles of L for the 
pencil. 

(2) The fourth harmonic of L for the pair of common chords 
through Pj. 

222. The locus of the centres of the pencil of conies is a conic 
which passes through 

(1) The points Pi, P^, Ps, 

(2) The mid-points of each of the six sides of the quadrangle, 

(3) The double points of the involution determined hy the 
pencil on the line at infinity \. 

Suppose L to be the line at infinity. 



Two conies inscribed in a quadrilateral. 

223. Fig. 102 shews us that in general two conies have four 
common tangents forming a circumscribing quadrilateral whose 
sides intersect iu six points. 

* Poncelet, Prop. Froj. Vol. i,Art. 373. 

t Dr Taylor, Geometry of Conies, p. 284 (1881), called the locus of centres 
"the eleven-point conic of the quadrilateral," but it seems better to apply 
the term to the reciprocal conic of L. 



I 




222-226] TANGENT VERTICES 211 

Def. The point of intersection of two common tangents we 
Tangent shall call a tangent vertex. 

vertex. rpj^^ g-^ tangent vertices will be denoted by the 

letter T with suffixes 1...6. 

As in Art. 209, we shall speak of any two opposite vertices of 
the circumscribing quadrilateral as a pair of tangent vertices. 

The figure shews us that there are in general three joairs of 
tangent vertices, viz. 1\, T.,; T^, T^; and 1\, T^. 

224. The distinctive property of a tangent vertex is that 

Any two lines through it which are conjugate for one of the 
conies are also conjugate for the other, 

for each is the harmonic conjugate of the other for the two 
common tangents through the vertex. Art. 166 (^'). 

225. The line joining a pair of tangent vertices T^, T^ has the 
same pole for both conies. 

For if it had two poles Q, Q' the line joining them would pass 
through T^, being the fourth harmonic of ^1:7^2 for the common 
tangents through Ty^. Similarly it would also pass through T^, 
which is impossible. 

226. Any straight line L which has the same pole I for both 
conies is a line joining a pair of tangent vertices. 

For let L meet one of the common tangents as T^l\ in T. 
Join Tl, and draw TQ, TQ' the other tangents from T to A and 
B. Then T(LIT^Q) is a harmonic pencil, as is also T (LIT^Q'), 
so that TQ' coincides with TQ, i.e. 7^ is a tangent vertex, and 
coincides with T^^ suppose. Similarly it may be shewn that L 
passes through T^. Hence 

The lines joining the three pairs of tangent vertices form a 
common self-conjugate triangle. 

14—2 



212 CROSS-KATIO GEOMETRY [CH. XIV 

Consequently, by Art. 212, 

The line joining a pair of chord vertices passes through a pair 
of tangent vertices, and similarly, the line joining a pair of tangent 
vertices passes through a pair of chord vertices. 

Def. a pair of common chords are said to correspond to 
Correspond- the pair of tangent vertices which lie on the polar 

ing common ^^f ^he vertex of the chords. 

chords and 

tangent Thus in Fig. 102 F^^g^g^ and P^g^g^, correspond 

vertices. to T^ and T^. 

227. The poles of any common chord lie on the line joining the 
corresponding tangent vertices. 

By Arts. 211, 226. 

228. If from any point P two pairs of tangents PQ, PR ; 
PQ', PR' are drawn to the conies, the pencil P (QR, Q'R', T^T^ ...) 
is in involution. 

By Art. 188. 

229. Tn Art. 228, if the point P is taken at one of the 
points of intersection of the conies, as g^, and if the tangents at 
^1 meet T^T^ in a, ^, the pencil g^ (a^T-^T^ is harmonic, g^a, g-^/S 
being the double rays of the involution pencil. 

Also by Art. 227 a, /? are the poles of the common chord ^1^4. 

230. By Art. 214 (aa', bb', pp) is an involution range, in 
which the double points are evidently P^ and P3. 

By Art, 228 Pi {aa, bb', T^T^ is an involution pencil, in which 
the double rays are PjPg ^^d P1P3. 

Therefore T^T^ is a pair of conjugate points in the involution 
range {aa!, bb' , pp). 

231. If three conies have a common tangent vertex, their three 
other corresponding tangent vertices are collinear. 

Let A, B, G be the conies, T the given tangent vertex, 
T^, T^, T^ the corresponding tangent vertices for A and B, 
B and C, A and C. 



i 



227-233] RANGE OF CONICS 213 

Draw ^i^2> ^^d from any point P on it draw pairs of tangents 
PQ, PR, &c. to the conies, and join PT, PT^. 

Then P (QB, Q'R\ Q"B") is an involution pencil, in which 
PT, PT^ ; PT, PT^ ; and PT, PT., are pairs of corresponding 
rays, i.e. T^ lies on the line ^1^2- 



Range of Conies. 

Range of • Def. A system of conies inscribed in a quadri- 

conics. lateral is called a range of conies. 

232. Given a range of conies, the pairs of tangents drawn to 
them from the same point P will form, a pencil in involution. 

See Art. 188. The double rays are the tangents at P to the 
conies which pass through P. 

By Arts. 225, 226 a range has one, and only one, common 
self-conjugate triangle. 

As in Art. 217 it may be shewn that 

A given pair of lines are conjugate for only one conic of a 
range. 

233. The poles of any straight line L for the range are 
collinear. 

Let l^, l^ be the poles of L for A and B, and let Ij^l^ meet 
L in P. 

Then PIa^b ^^^^ ^^ ^^^ conjugate lines for both A and B, and 
are therefore the double rays of the involution pencil formed by 
the pairs of tangents from P with the lines drawn from P to 
opposite vertices of the quadrilateral. But the tangents from 
P to a third conic C of the range belong to the same involu- 
tion, and consequently have the same double rays, viz. their 
harmonic conjugates. Hence PZ^Z^ passes through l^, the pole of 
L for G. Hence 



214 CROSS-RATIO GEOMETRY [CH. XV 

A pair of lines which are conjugate Jbr two conies A, B are 
conjugate for the range to which A and B belong. 

If OL, OL' are a pair of conjugate lines for a range, it is 
evident that is a common point of the two conies of the range 
which touch the lines OZ, OL' respectively. 

The line L and the locus of its poles being conjugate lines for 
the range we infer that 

All conies touching three given lines and having a given pair of 
lines conjugate touch a fourth line. 

Hence, if three tangents are given and a pair of conjugate 
lines, the conies touch a definite fourth line, viz. the fourth 
common tangent of the two conies which touch the three given 
tangents, and of which each touches one of the two conjugate 
lines at their intersection. 

In other words : 

If three tangents are given to a conic, a pair of conjugate lines 
are equivalent to a fourth tangent. 

We leave it to the student to shew as in Art. 218 that 

(1) A system of conies touching two give7i lines and having 
given two pairs of conjugate lines constitute a range, and 

(2) Two tangents and three pairs of conjugate lines determine 
a conic. 

234. In Art. 233 if the line L rotates about a fixed point P, 
the line which is the locus of the poles of L will envelop a conic 
which also touches the polars of P for the range. 

By Art. 233 it is only necessary to consider two of the conies, 
A and B. Let P^, Pj^ be the polars of P for A and B. Then 
if l^ and l^ are the poles of L in any position, ^^ will lie on P^ , 
and l^ on P^ ; and as L rotates about P, the range of polars (L) 
is homographie to each of the ranges of poles (Z^) and {Ij^), by 
Art. 167. 



234-237] ELEVEN-TANGENT CONIC 215 

Therefore the range {I a) = the range (l^), by Art. 39, and by- 
Art. 139 the line Ij^l^ envelops a conic touching the lines Fj^ 
and F^. 

The same reasoning shews that the conic touches the polar of 
F for each conic of the range. 

235. The cross-ratio of the range formed by the poles of any 
line L for four conies of a range is independent of the position 
of L. 

By Art. 233 the poles of Z, viz. l^^ /^, l^^ Ij), lie on a line X', 
and if M is any other line its poles, viz. m^, m^, m^., rrijy, lie on 
another line M'. Let L and M meet in the point F. Then the 
polar of P for A is the line l^'^Ai by Art. 163, and similarly 
for its polars for the other conies. By Art. 234 these polars are 
tangents to a conic which touches L' and M'. 

Therefore by Art. 130 the range {lAhh^D) = {'^A'"^B''^c^D)i 
i.e. the cross-ratio of the range formed by the poles of L is equal 
to that of the range formed by the poles of any other line. 

Note. Chasles called this the cross-ratio of the four conies of 
the range. 

236. When the line L passes through F^, its poles lie on the 
line F2P3, and the conic envelope touches F^F^. Hence 

The conic envelope touches the sides of the common self conjugate 
triangle. 

237. The conic envelope of L for the range touches the 
following eleven lines : 

(1) The three sides of the common self-conjugate triangle. 

(2) The six lines obtained by joining any tangent vertex to 
the point P, and taking the harmonic conjugate of this line for 
the two common tangents through that vertex. 

(3) The double rays of the involution pencil obtained by 
drawing pairs of tangents from F to each conic of the range. 



216 



CROSS-RATIO GEOMETRY" 



[CH. XIV 



Hence the conic envelope of L may be called "the eleven- 
tangent conic." 

Cor. If P lies on one of the sides of the common self-con- 
jugate triangle as P^Pz^ the conic envelope degenerates into two 
points, viz. 

(1) The point P^, through which the polars of P pass, and 

(2) The fourth harmonic of P for the jmir of tangent 
vertices on P^P^- 

238. The locus of the centres of the range is a straight line*. 
Suppose the line L in Art. 233 to be at infinity. 

239. Let PQRS he a quadrangle, A, JB, C its diagonal points, 
and let X, Y he two conjugate points for the pencil of conies cir- 
cumscribing the quadrangle, 

A 




Fig. 103. 
Newton, Princip. Bk i, § v, Lemma 25, cor. 3. 



^M 238-240] Dr w. p. milne's theorem and correlative 217 

H Then X (PQRS) = Y{XABC)*. 

^f ^^ the pencil of conies, if we draw the two a, /3 passing 
respectively through the points X, Y, they will have the line 
XY for a common tangent by Art. 218. Then T, the pole of 
PS for a, lies on AB, by Art. 216. Let PS meet XY in F, XT 
in G, and AB in H, Then F is the pole of XT for a, and 
therefore, by Art. 166 (a), F and G are conjugate points ...(1). 
Then the conic-pencil 

X{PQPS) = P{PQPS)--={TABH) (2). 

Now by Art. 221 the eleven-point conic which is the locus of 
the poles of the line XY for the pencil of conies passes through 
the points A, B, C, X, Y, and also through the point G, since 
(FGPS) is harmonic, by (1). 

.*. the conic-pencil 
Y{XABC) = G (XABC) = {TABH) = X(PQES) by (2). 

240. Let PQRS be a quadrilateral, ABC its diagonal triangle, 
and let Ox, Oy he two lines conjugate for the range of conies 
inscribed in the quadrilateral. Let one of them. Ox, meet the lines 
PQ, QR, RS, SP in the points p, q, r, s, and let the other, Oy, 
meet BC, CA, AB in the points a, ft, y. 

Then (pqrs) = {Oa/3y). (Correlative of Dr Milne's Theorem.) 

Of the range of conies, if we draw the two a, b touching 
respectively the conjugate lines Ox, Oy they will have for a 
common point. Let a touch the lines PQ, QR, RS, SP in the 
points Op, aq, a^., ag. 

Then the polar of P for a passes through C. Let this polar 
meet Ox in x. Then since P is the pole of Ca^, and is the 
pole of Ox, therefore x is the pole of PO, 

Therefore Px, PO are conjugate lines for a ( 1 ). 

* This theorem, with a proof by analysis, was given by Dr W. P. Milne 
in the Math. Gazette, Jan. 1911, p. 386. 



218 



CROSS-RATIO GEOMETRY 




.*. the range {pqrs) on the tangent Ox 

= conic-pencil of points of contact {a^a^fb^a^ Art. 13L 

= the range on the tangent FQ 

= {a,QTP) (2). 

Now by Art. 237 the eleven- tangent conic, which is the 
envelope of the polars of for the range of conies, touches the 
lines BC, CA, AB, Ox, Oy, and also touches the line Fx, since 
F (xOQS) is harmonic, by (1). 

Let Fx meet BC in k and GA in I. 

Then the ranges made by the four tangents Ox, BC, CA, AB 
on the two tangents Oi/ and Fx are equicross, i.e. 

{Oa/3y) = {xklP)=C{xklF) = (apQTF) = (pqrs) by (2). 




I 



EXAMPLES 219 



EXAMPLES. 

1. If two conies touch one another, the line joining the poles of their 
common chord passes through the point of contact. 

2. Two conies touch one another at P, and T, T' are the poles of their 
common chord IF. If any chord PQR is drawn through P, TR, T'Q will 
intersect on II'. 

Conversely, if TR, T'Q intersect on II\ QR passes through P. (Chap. 
XIX, Ex. 28.) 

3. A system of conies is described touching a given conic at a given 
point P, and intersecting it in two fixed points I, I'. The locus of the pole 
of ir for the system is a straight line passing through P. (Chap. XIX, 
Ex. 29.) 

4. A system of conies which pass through three given points and have 
the poles of the line joining two of them on a fixed line have a fourth 
common point. 

5. Given two tangents CA, CB, and two points J, Z' on a conic, the 
locus of the pole of the common chord II' is a double line of the involution 
pencil C{ir, AB). (Chap. XIX, Ex. 30.) 

6. Two conies intersect in the points A, B, I, I'. Any chord through 
A meets the conies in P, Q. Then B (II'PQ) is constant. (Chap. XIX, 
Ex. 33.) 

[In Examples 7 — 12 the reference is to Fig. 102.] 

7. P, Q are the points of contact of one of the common tangents T^T^. 
The pencil P^ {PQg^g^) is harmonic. Art. 213. 

8. If the tangent P-^a meets the pair of common chords through Pg 
in G, G', and the conic B in H, K, the ranges {PiaGG') and {P^aHK) are 
harmonic. 

9. The tangent to the conic B at any point Q meets A\xi R, S, g^g^ in 
(r, and ^3^4 in G'. Then P^Q is one of the double rays of the involution 
pencil Pi (RS, g^g^. Art. 213. 

10. If P, Q are the points of contact of the common tangent T^T^, the 
six points P, Q, P-^, P^, gi, g^ lie on a conic having PQ, g-^g^ for conjugate 
lines. See Ex. 7. 

Also P, Q are conjugate points for any conic of the pencil through 
gig29394- By Desargues. 



220 CROSS-RATIO GEOMETRY [CH. XIV 

11. A straight line through T^ meets the conic A in a, a' and B\ia.h,h' 
so that the points are in the order T^aba'h', Then aa, j86 meet on g-^g^^ as 
do also aa', /36'. 

12. If the pole of g^g^ for B lies on A, the pole of g^^ for B will also lie 
on A. 

13. Three conies have a common chord the poles of which are collinear. 
If the polars of a point for the conies are concurrent the conies belong to a 
pencil. 

14. A pencil of conies passes through the points A, B, I, I', and is a 
fixed point in the plane. For any one of the conies T is the pole of II', and 
the polar of G meets TC in P. Shew that the locus of P for the pencil is a 
conic through G, I, I'. 

15. In Art. 218, by taking the pair of conjugate points to be the 
circular points i, i', shew that 

(1) If two conies of a pencil are rectangular hyperbolas, so are all the 
conies of the pencil. 

(2) Two conies of the pencil touch the line at infinity at the points i, i' 
respectively. 

(3) li A, B, G are three of the common points, the locus of the centres 
is the nine-point circle of the triangle ABC. 

16. A is one of the common points of a pencil of conies. Any trans- 
versal through A meets three of the conies in P, Q, R, and a common chord 
in K. As the transversal rotates about ^, (PQEK) is constant, by Desargues. 

17. II' is a common chord of two conies. Through T, T' its poles are 
drawn two chords intersecting on 11'. Shew that each of the lines joining 
the extremities of the chords passes through a tangent vertex. 

18. Shew that in Fig. 102 the points T^, 1\ , gi, g2^ ds, O4 lie on a conic, 
i.e. that one of the conies of a pencil passes through a pair of tangent vertices 
of each pair of the conies. 

19. In Fig. 102 shew that a, /3 are conjugate points for the conic of the 
pencil which passes through the points Ti, T^. 

20. Two conies touch at G and have a common chord II'. The tangent 
to one conic a at ^ meets the other conic /3 in B, B', and T being the pole of 
II' for a, TA cuts a in A'. Then GA' is one of the double rays of the 
involution pencil G {BB', II'). 




I 



. EXAMPLES 221 

21. If four conies are inscribed in a quadrilateral, the cross-ratio of the 
poles of any straight line is equal to the cross-ratio of the points of contact 
on each of the sides of the quadrilateral. 

22. A given line L meets any conic of a pencil in a, a'. Prove that a, a' 
are conjugate points for the eleven-point conic corresponding to the line L. 

23. If a conic y is contra-polar to each of two conies a, /3 : 

(1) 7 will have a chord 11' common to a and j3, 

(2) The pole of II' for y will lie on AB, the other common chord of 
a and /3, 

(3) A and B are conjugate points for y, 

(4) 7 will be contra-polar to the pencil of conies through A, B, I, I', 

(5) 7 will pass through two of the vertices of the common self- 
con jugate triangle of the pencil. 

24. In Fig. 102 shew that any conic through gig^a^ is contra-polar to the 
conic of the pencil which passes through T-^, Tg. See Ex. 18. 

25. In Fig. 102, g^a, g^ meet the conies A^ B in a^, b^. Shew that the 
conic round T^T^g-^^g^g^g^ is contra-polar to the conic round gig^g^cLx^i- 

26. A variable conic passes through a fixed point and intersects a conic 
to which it is contra-polar in two fixed points. Shew that it passes through 
a fourth fixed point. 

27. Three given conies have a common chord. Shew that the locus of 
a point whose polars for them are concurrent is a conic contra-polar to each 
of the given conies. 



CHAPTER XV 
HOMOLOGY 

THE HOMOLOGUE OF A LINE AND CONIC. RELATIONS 
BETWEEN A PAIR OF COMMON CHORDS AND THE 
CORRESPONDING PAIR OF TANGENT VERTICES. RELA- 
TIONS BETWEEN THE FOUR CONSTANTS OF HOMOLOGY 

241. Def. Given a fixed point T and a fixed line X, and any- 
plane figure A, if through T we draw a transversal 
meeting ^ in P and L in G, and if on the trans- 
versal TP we take a point P' such that the cross- 
ratio (TGPP') is constant {=X.'), then 

The locus of the point P' is called the homo- 
logue of A, 
The point T is called the centre of homology ^ 
The line L is called the axis of homology^ 
A.' is called the constant of homology. 

242. The homologue of a straight line is a straight line. 



Homologue. 
Centre of 
homology. 
Axis of 
homology. 
Constant of 
homology. 




241-243] HOMOLOGUE OF A LINE AND CONIC 



223 



In the above Def. let ^ be a straight line, and let it meet L 
in g. Through T draw two transversals, one meeting A in. P and 
L in G, the other meeting A in P-^ and L in G'. On TP and 
TP^ take points P' and P/ such that 

\'^{TGPP') = {TG'P,P^'), by Art. 9. 

Then by Art. 41 (2) these ranges, being homographic and 
having T a common point, are in perspective, and consequently 
P'Pj passes through the fixed point g. Therefore the homologue 
of ^ is a straight line passing through g. 

If the line A is at infinity, its homologue is evidently a line 
parallel to L. 

243. The homologue of a conic is a conic. 




Lemma. In Art. 74 we proved that if abc..., a'b'c'... are two 
homographic ranges on the same straight line, and e, f their 
common points, the cross-ratio (aea'f) is constant, where {a, a') 
are any pair of corresponding points. Conversely, if e^fsiYei two 
fixed points on a straight line, and abc..., a'b'c ... two rows of 
points on the same line such that (aea'f) is constant, (a, a') being 
any pair of corresponding points, the two rows are homographic 
and e, / are their common points. 



224 



CROSS-RATIO GEOMETRY 



[CH. XV 



Through one of the common points f draw any straight line 
Z, and from any point g on it draw rays to the points of the rows, 
and from e draw a transversal M meeting these rays in the points 
aj^iCi..., CTi'^/c/ . . . , and meeting L iny^. 

Then by Art. 41 (1) the systems on the two transversals 
through e are homographic, being in perspective, centre g. Also 
by Art. 39 the rows a^h-^c-^...^ a^h-[c-[... are homographic, and e^f-^ 
are their common points. 




Fig. 107. 

Now let ^ be a conic, L the axis of homology cutting A 
in ^, g\ and T the centre of homology, and let if be a fixed 
transversal through T. 

Through T draw any transversal meeting A in P, and L in (r, 
and on this transversal TPG take a point P' such that 

{TGPF) = a constant X', by Art. 9. 
Then we will prove that as the transversal rotates about T, the 
locus of P' is a conic passing through the points g, g . 




243-244] HOMOLOGUE OF A CONIC 225 

Through T draw any other transversal meeting the conic A in 
Pj, and X in 6?!, and on it take the point P/ such that 

Then by Art. 41 (2) the two ranges {TGPF) and (TG^P^P,'), 
their cross-ratios being equal, and their point of intersection T 
being a common point, are in perspective, the centre of per. 
spective lying on the line L. 

Suppose now we draw a system of transversals through T, 
and form on each a range = k' as above. Join g to the two series 
of points PP^P^..., P'P{P^.... Also join gT, and let the pencil, 
centre ^, which is thus formed be cut by the transversal M in 
the points a^\c^...^ a^h(c{ ..., T^ f^. 

Then by Art. 40 (2) X = {Tf^a,a^) = (Tf^hh^) = .... 

Therefore by the above Lemma a-^b^Cj^ ... and a^hiCi... are 
two homographic rows of which T,f^ are the common points. 

.*. (ai6iCi...) = «6/ci'...), 

.-. g{(hhiCi...) = g{aj'biC,' ...), 

:.g{PP,P,...)=g{FP;P^...). 

In a similar manner we can shew that 

g'{PP,P,...)=g'{P'P^P^...). 

But g{PP,P,...)=g'{PP,P,...), by Art. 129. 

.-. g(P'P,'P,' ...) = g'{P'P,'P,' ...), by Art. 44. 

Therefore by Art. 138 the locus of P' is a conic passing 
through g and g'. We will call this conic £. 

244. It will be noticed that in the course of Art. 243 we 
incidentally proved the following property : 

If through the centre of homology we draw two transversals 
TPP' and TP^P(^ each meeting the conies in /our points PP'pp' 
and PiPiPiPi, the lines joining corresponding pairs of these points 
such as PPi, P'Pj will meet on the axis of homology. 

M. 15 



226 CROSS-RATIO GEOMETRY [CH. XV 

If the transversal TP^ rotate into the position TP^ the lines 
P^P and P^P' become the tangents at P and P', which therefore 
intersect on gg'. 

If the transversal through T touches ^ at ^ and meets B in 
Q' and L in F', then since the tangents at Q and Q' intersect on 
L, viz. at F\ it follows that the tangent at Q is also the tangent 
at Q'. Hence T is the point of intersection of a pair of common 
tangents and gg' is one of the corresponding common chords. 
Consequently if we have two conies, one of them is the homo- 
logue of the other, a tangent vertex being the centre of homo- 
logy, and one of the corresponding common chords the axis of 
homology. 

Relations between a pair of common chords and the 
corresponding pair of tangent vertices. 

245. The object of the next two articles is to prove the 
following propositions: 

(1) If from any point on a common chord we draw the four 
tangents to the two conies, the straight lines joining the points of 
contact on the first conic to the points of contact on the second conic 
will pass hy pairs through the two tangent vertices which correspond 
to the common chord. 

(2) If through a tangent vertex we draw any transversal 
meeting the conies in four points, and draw the tangents at these 
points, the pair of tangents to the one conic nnll meet the tangents 
to the other in four points which lie hy pairs on the pair of common 
chords corresponding to the tangent vertex. 

246. If from, any point on the a^s of homology four tangents 
are drawn, one pair of the lines joining the points of contact pass 
through one of the corresponding tangent vertices, and another pair 
pass through the other. 



1 



245-246] COMMON CHORDS AND TANGENT VERTICES 227 



qVX 



pp 



n'l 



Fig. 108. 



15-2 



228 CROSS-RATIO GEOMETRY [CH. XV 

If through T^ we draw any chord meeting the conic A inp 
and gg' in F, it will meet the conic B in two points q^ q\ and of 
these points one {q suppose) will be such that (T^Fpq) = X'^ and 
the tangents at p, q will intersect in a point n which lies on gg', 
by Art. 244. Conversely, if on gg' we take a point n, and draw 
the four tangents np, nq, npi, nq^, the line joining the points of 
contact of two of these will pass through T^ . Let p, q be the 
pair. The line joining another pair of points of contact will also 
pass through I^j, for if we join T^ to either of the points pi or q^, 
it will obviously pass through the other, for the tangents at these 
points intersect in n. 

We will now shew that the line joining the points of contact 
Pi Qi passes through 7^2* 

The lines pp^ and qq^ are the polars of n, and intersect in a 
point m which lies on gg', by Art. 210, and they pass respectively 
through the fixed points a, (i', which are the poles of gg', and lie 
on T^l\, by Art. 227. 

Since the triangle ma'/3' is cut by the transversal T^pq, by^ 
Menelaus' Theorem 

mp oTj ^ 

a!p' (i'T^'mq ^ '' 

Since (ap^mp) is a harmonic range, 

mp mpi 
ap~~ a'pi' 

Since (T^T^a/B') is harmonic by Art. 229, 

a'T,_ a'T^ 
P'T, PT,' 

Therefore by substitution in (1) we obtain 

mp, oT^ §^^i 
d'p, ' /3'T^ ' mq 

Therefore, by the converse of Menelaus' Theorem, p^^q passes 
through T^. Similarly q^p passes through T^, 




246-248] COMMON chords and tangent vertices 229 

We have thus proved that certain relations hold between a 
pair of tangent vertices and one of the corresponding common 
chords, and it could be shewn in exactly the same manner that 
the same relations hold for the same pair of tangent vertices and 
the other corresponding common chord, but of course the constant 
of homology would have a different value. 

Hence, if we take any point on one of a pair of common chords, 
and from, it draw the four tangents to the two conies, and take any 
pair of the points of contact which lie on different conies, then of 
the four lines which join them two will pass through one of the pair 
of tangent vertices corresponding to the common chords, and the 
other two will pass through the other tangent vertex. 

247. If from a tangent vertex we draw any transversal 
cutting the two conies in four real points, and draw the tangents 
at these points, the tangents to one conic will meet the tangents to 
the other in four points which lie hy pairs on the two common 
chords corresponding to the tangent vertex. 

For let the transversal through T^ cut the conies in p, q, and 
let the tangent to ^ at jt? meet the common chords in n, n^. 
Then by Art. 246 of the two tangents from n to B one of the 
points of contact lies on T^p and is therefore q, i.e. the tangent at 
p meets the tangent at q on the common chord gg'. Similarly 
it may be shewn that the other points of intersection of the 
tangents at p, q, p', q' lie on one or other of the pair of common 
chords corresponding to T^. 

In Fig. 108 the four points are n, n^, n', n/. 

248. From Art. 246 we see that in Fig. 108 

If m is any point on a common chord Pg, poles a, /B', and 
ma, m^' meet the conies in two pairs of points p,Pi, q, qi', the lines 
joining the pairs of these points which lie on separate conies will 
pass by pairs through the points T^, T^ corresponding to the common 
chord Pg. 

Also X' = {T^Fpq) = {T^c'a'P), by Art. 41 (1). 



230 CROSS-RATIO GEOMETRY [CH. XV 

Relations between the four constants of homology. 

249. In Fig. 108 let T-^pp'qq be any transversal through 1\ 
We have shewn in Art. 247 that the tangents at jo, q intersect 
on the common chord PF^ as do also the tangents at p', q'^ i.e. T^ 
is a centre and PF the corresponding axis of homology of the two 
curves, the constant of homology being \' suppose. Also by 
Art. 247 the tangents at p, q' intersect on the common chord Pf, 
as do also the tangents at p\ q\ i.e. T^ is a centre and Pf the 
corresponding axis of homology of the two curves, the constant 
of homology being A. suppose. 

Similarly T^ is a centre of homology, and has Pf and PF as 
its corresponding axes according as we take for constants /a or /a'. 

By Art. 248 these four constants of homology are determined 
by the equations 

{T,cal3) = K {T,c'a'ft') = X', 

Writing X in the form ^^ : -^ , &c. we see that these equa- 
1 ip cp 

tions are connected by the relations 

Similarly ^, =^ (T^T^a' P') = - 1. 

Also, if T^QGQ'G' is a common tangent, 

\ = {TmQ'\ X'={T,G'QQ'). 

'' X' = §^' ^ ^ = (^'^W) = -1, by Art. 213, Cor. 



X = -\' = 



fX = fX 



250. The results obtained in the previous article may be 
stated as follows : 

Considering two conies as homologous figures, if we take one of 
a pair of tangent 'vertices as centre of honfiology, and one of the 



249-251] THE FOUR CONSTANTS OF HOMOLOGY 231 

pair of corresponding common chords as axis, the constant of 
homology has the same value both in magnitude and sign as when 
we take the second tangent vertex as centre of hom,ology and the 
second corresponding common chord as axis ; and it has its value 
the same in magnitude hut of opposite sign when we take the first 
vertex as centre with the second chord as axis, or the second vertex 
as centre with the first chord as axis. 

251. In Figs. 107, 108 let PG intersect A and B in the 
points g^, g^. Then retaining the same centre T^, axis PG, and 
constant X, if we construct the homologue of B we shall obtain a 
third conic C, which will touch the common tangents from T^ to 
A and B, and pass through the points ^i, g^^; but the second 
centre and axis will not be T^ and PG'. Similarly a fourth conic 
can be obtained from C, and so on. This system is a particular 
case of that considered in Art. 288. 

If X = — 1, the homologue of B will be A, and the system 
Harmonic will in this case consist of these two conies only, 

homology. For (T^GPF) - - 1 = (T^GP'P). This is called 

harmonic homology*. 

Also since (T^c'a'ft') = 1, T^ coincides with c', and therefore 
the centre T^ lies on the axis PG'. Similarly T^ lies on the 
axis PG. 

EXAMPLES. 

1. Shew that if the conies described in Art. 251 are denoted by 
A, J5i, B2, B3... the successive conies ^1, B2, B^... are the homologue s 
of A for the constants X, X^, X^ ... respectively. 

2. In the general case shew that the second centres for any pairs of the 
system are collinear, and that all the second axes are concurrent. 

3. If with the same centre and axis, and constants X, -X respectively, 
two conies jBj , B2 are homologues of A , shew that Bi and B2 are in harmonic 
homology. 

* Eussell, Elementary Treatise on Pure Geometry, p. 325, 



CHAPTER XVI 

CONSTRUCTION OF COMMON CHORDS AND TANGENT VERTICES 
AND COMMON SELF-CONJUGATE TRIANGLE OF TWO 
CONICS 

252. We will now consider the reality and imaginarity of 
the different groups each consisting of a pair of common chords 
and the corresponding pair of tangent vertices, and we will shew 
how to construct them when it is possible to do so. 

253. Poncelet in his Prop. Proj. Vol. i, Art. 54 divides 
common chords into three classes : 

(1) real, when they pass through two real points of inter- 
section of the curves, 

(2) ideal, when the points of intersection of the curves 
through which they pass are unreal, but the chords themselves 
are real, 

(3) imaginary, when the chords cannot be constructed. 
Similarly, tangent vertices may be 

(1) real, when they are the intersection of two real common 
tangents, 

(2) ideal, when the tangents through them are unreal, but 
the points themselves are real, 

(3) imaginary, when the points cannot be constructed. 



1 



i 



252-255] CONSTRUCTION OF COMMON CHORDS, ETC. 233 

^f 254. I. When the conies intersect in four real separate 
points. 

iln Fig. 102 the three groups 
i Tu T^y 9x9^, 9i9^ (1)» 
^ ^3, ^4, 9x9^. 9^9^ (2), 
^^ ^6, ^6, 9i9zy 9i9A (3), 
are all real, as are also the three vertices Pj, Pg, Pg of the 
common self -conjugate triangle. 

255. II. When two of the points^ as 9^, ^4, coincide, whilst 
the other two, 9^, ^3, are real and separate. 




Fig. 109. 



T3=Tfi 



Here g^=g^= T, = F, = P,, 

T, = T, and T,= T,. 
The three groups are all real, but (2) coincides with (3). 



234 



CROSS-RATIO GEOMETRY 



[CH. XVI 



256. III. When the conies have double contact along the line 
LMj N being the intersection c^ tangents at L, M. 




M 
Fig. 110. 

Pj and Pg are indeterminate, being any pair of conjugate 
points on LM. 

The three groups are all real, being 
M, L, LN, MN, 

N, N, LM, LM, 
N, iV^, LM, LM, 
i.e. (3) coincides with (2). 

257. lY. When the conies osculate at the point L the three 
groups are real and identical, being N, L, LM. 

L = g, = g^ = g,= T, = T,= T, = P, = P,=.P„ 



N=T.= T, 



n. 



258. V. When the conies have four consecutive points common. 
The three groups are all real and identical, the common chords 

being the common tangent, and the tangent vertices all coinciding 
at the point of contact. 

259. VI. When the conies intersect in only two real points. 



256-259] CONSTRUCTION OF COMMON CHORDS, ETC. 235 




236 CROSS-RATIO GEOMETRY [CH. XVI 

First method. 

Let the two real common tangents intersect in ^4, through 
which draw a transversal meeting the conies in the four real points 
a, a\ b, h'. Let the tangents at a, h meet in n, and let them 
intersect the real common chord g^^g^ in m, m'. Then by Art. 247 
ma', Tn'h' are the tangents at a', h' ; produce them to meet in n'. 

By Art. 247 n, n' are two points on the corresponding 
common chord, which is ideal. Join nn\ meeting ^1^2 in P^ 
which has one common polar for the two conies. Draw the 
tangent mb" and also the common polar of Pj, which will pass 
through ^4 by Art. 226, and let it meet h"a' in T^. Then 
by Art. 246 T^ is the real point of intersection of a pair of 
imaginary common tangents, and is an ideal tangent vertex. 

Let the polar of P^ meet the conies in %, cf,^, b^, 6/. By 
Art. 230 P1P2, P1P3 are the double rays of the overlapping 
involution P^ia^a^, b-Jb-^), and are consequently unreal. 

Therefore the points Pg, P3 are imaginary, and lie on the 
real line T^T^. 

A little consideration will shew us that the other two 
pairs of common chords must be imaginary. For by Def. of 
Art. 209, two common chords which do not belong to the same 
pair must intersect at a point where the curves also intersect. 
Consequently, if it were possible to construct a second pair of 
common chords, they would meet the pair already drawn in 
points where the curves intersect, which is contrary to the 
hypothesis that the curves intersect in only two real points. 

Second method. 

Draw any transversal meeting the curves in a, a and )8, /8', 
and the real common chord in y, and find y such that the 
range (aa, /8/8', yy') is in involution, Arts. 104, 108 ad fin. 
Then by Art. 213, y is a point on the second common chord. 

Similarly, by drawing any other transversal we can find a 
second point y" on the second common chord. The line joining 
y'y" is the line required. 



J 



I 



259-260] CONSTRUCTION OF COMMON CHORDS, ETC. 237 

Third method. 

Let the common tangent TJPQ meet the real common chord in 
(r, and on it find the point G' such that the range {PQGG') = — 1, 
Arts. 29, 32, Oor. Then, by Art. 213, Cor., G' is a point on 
the second common chord. 

Similarly from the other common tangent we can find another 
point G" on the second common chord. 



260. VII. When the two conies touch externally. 



T,=T. 



T, 



Fig. 113. 



238 



CROSS-RATIO GEOMETRY 



[CH. X^ 



The common tangent at the point of contact T^ is also a real 
common chord. The line ^37^4 is a common polar by Art. 225, 
and its pole P^ is the point of intersection of a pair of common 
chords. Through T^ draw any transversal meeting the conies 
in a, h. If the tangents at a, h meet in n, the line P^n is the 
second common chord (ideal) by Art. 247. 

The three groups consist of 

(2) T,, T,, P,T,, P,n, 

(1) and (3) T^, T^^ and two imaginary lines intersecting in the 
real point T^. 



261. VIII. When the conies touch internally. 




\ 



Fig. 114. 

The tangent at T^ is a real common chord. Through T^ draw 
any transversal T^ah meeting the conies in a, h. By Art. 247 the 
tangents at a, h meet at a point n on the second common chord. 



261-262] CONSTRUCTION OF COMMON CHORDS, ETC. 239 

Through n draw the other two tangents, touching the conies in 
a\ h'. Let ab\ a'b meet in T^ (an ideal tangent vertex). Let P^ 
be the pole of T^T^. Then P^n is the second common chord 
(ideal). P^ and P^ coincide at T^. 

262. IX. When the conies have no real points of intersection^ 
each conic being entirely without the other. 




Fig. 115. 



240 



CROSS-RATIO GEOMETRY 



[CH. X\ 



Let a transversal through T^ cut the conies in aa\ bb\ Let 
the tangents at a, b' meet in m, and those at a, b in n, and find 
Pg the pole of ^^1^2 with respect to either conic. Then by 
Art. 247 Pa^, ^2^^ ^^^ ^ joair of ideal common chords. The 
other common chords are imaginary, one pair intersecting in 
the real point Pj, the other pair in the real point Pg. 

263. X. When the conies have no real points qf intersection^ 
one being entirely within the other. 




263] COMMON CHORDS AND TANGENT VERTICES 241 

To solve the problem in this case we will make use of the 
property of Art. 219, viz. if we have a straight line Z, and if P' 
be the intersection of the polars of any point P on Z, then as P 
moves along L, P' describes the eleven-point conic passing through 
the poles of L for A and B. 

Denote the reciprocal conic by C, and describe it by points, 
denoting the different points on L by the numbers 1, 2, 3...n..., 
and the points on C corresponding to them by 1', 2', 3'...n'.... 

Take another line M, and construct its reciprocal conic C", 
denoting the different points on M by the letters a, 6, c . . . m . . . , 
and the reciprocal points on C by a', h', c ...m! .... 

Then it is clear that if L arid M intersect in R^ its reciprocal 
point R' will be common to C and C. 

Let Pi, Pgj ^3 be the other three points of intersection of C 
and C. These will always be real, except in Case VI, Art. 259, 
where A and B have only two real points of intersection, when 
only one of the points Pj, Pg, P3 Avill be real. 

Suppose we consider one of the points as Pj. Regarded as 
a point on C, the two polars of P^ intersect at a point on L, and 
as a point on C they intersect at a point on M. Consequently 
they must coincide, and hence Pj is a point which has the same 
polar for A and P, and similarly for P^ and P3. Therefore, by 
Art. 212, P^PJP^ is the common self -con jugate triangle of A 
and B. 

If w, n' denote any pair of reciprocal points, they are con- 
jugate points, by Art. 219, the polar of n for the pair of common 
chords through Pj (say), which is one of the conies of the system, 
passes through n', and the rays Pjn, P^n! form a harmonic pencil 
with the common chords through Pj, Art. 121, Def. 

The same propert}}. holds for all pairs of reciprocal points 
(w, vi). Also Pg and P3 are conjugate points. Therefore, giving 
to n the different values 1, 2, 3..., by Art. Ill, 

Pi(n', 22', ^Z'...nn'...P^.^ 
forms a pencil in involution in which the double rays are the 

M. 16 



242 



CROSS-RATIO GEOMETRY 



[CH. XVI 



common chords through Pj, and similarly for the common chords 
through Pa and P3. 

Now the pair of common chords through P3 is clearly imagi- 
nary. And if we give to n any value such as 5 or 8, we see 
from the figure that the pencil P^{P^P^, nn') is non-overlapping, 
whilst P^i^P^P^, nn) is overlapping. Consequently the common 
chords through P^ are real, and those through P^ are imaginary. 

It is obvious from the figure that one of the double rays 
through Pj, viz. P^G, cuts L between the points 1 and 2, near 2, 
and the other, P^G\ cuts it between 5 and 6, at about 5*7. Hence 
P^G and P^G' are a pair of ideal common chords. 

P2 




Fig. 117. 

To find the tangent vertices take any point m on one of the 
common chords P^G^ draw the tangents ina^ 7na\ mb, mh' . Then, 
by Art. 246, ah^ ah will intersect P2P3 in the points T^^ T^^ a 
pair of ideal tangent vertices. 



CHAPTER XVII 

CONICS HAVING DOUBLE CONTACT 

T 




Fig. 118. 

264. If in Fig. 102 the points g^ and g^ move up to and 

Conies having coincide with one another, as also the points g., 

and ^4, the two conies will touch one another at 

the points Q, E, as shewn in Fig. 118, and are 

said to have double contact with each other. 

16—2 



Double Con 
tact. 



I 



244 CROSS-RATIO GEOMETRY [CH. XVIl 

The pair of common chords g^^, g^^ coincide in the hne QR^ 

„^ ^ ^ which is called the chord of contact, and the corre- 

Chord of . 

Contact. sponding pair of tangent vertices T^, T^ coincide in 

l*ole of the point T, which is called the pole of contact. 

The pair of common chords g^g^ and g^^, will 

also coincide with QR, and their corresponding pair of tangent 

vertices T^, T^ coincide with T. 

Also the tangents TQ, TR are the limiting positions of the 

jyair of common chords g-^g^, g^^, of which the corresponding 

2)air of tangent vertices 7\, T^ are at Q, R respectively. 

265. Any two conjugate points on the chord of contact form 
with the pole of contact a common self-conjugate triangle. 

For if Fj G are the two conjugate points, FT is the polar 
of G, GT is the polar of F, and T is the pole of GF. Hence 

Any point on the chord of contact has the same polar for both 
conies, and this polar passes through the pole of contact. 

In other words : 

If any transversal is drawn through the pole of contact, the 
ta/ngents at the points where it cuts the conies all pass through the 
same point on the chord of contact. 

This also follows from Art. 247. 

266. If any transversal meets the conies in aa', and hh\ and 
the chord of contact in F, the point F, by Art. 187, is obviously 
a double point of the involution range whose characteristic is 
{aa', hh'). Hence 

Any tangent to the one is cut harmonically at its point of 
contact, and at the points where it meets the chord of contact and 
the other conic. 

267. The polars of any point E intersect on the chord of 
contact. 



265-271] CONICS HAVING DOUBLE CONTACT 245 

Let the polar of E for the conic A meet QR in F, and let 
^^meet the conies A and B in aa, and hb'. Then by Art. 187 
F is one double point of the involution range given by {aa, bb'), 
and F, its harmonic conjugate for a, a', is obviously the other. 
Hence the polar of F for £ passes through F. 

Cor. Of two conjugate points one is always on the chord of 
contact. 

268. Again, if from any point E we draw pairs of tangents 
to A and B, the line FT is a double ray of the involution pencil 
determined by these pairs of tangents by Art. 228, since a pair 
of tangent vertices coincide at T. 

269. The poles of any straight line L are collinear with the 
pole of contact. 

Let l^j l^ be the poles, and let the line joining them meet 
L in P, and from P draw the pairs of tangents to A and B. 
Then by Art. 233 PIJ^b is a double ray of the involution pencil 
determined by these tangents, and therefore, by Art. 188, passes 
through T. 

The two lines L and IJ^^ ^^® conjugate, hence 

Of ttvo conjugate lines one always passes through the pole of 
contact. 

270. If a common chord of two conies has the same pole, the 
conies have double contact along the chord, and if a tangent vertex 
of two conies has the same polar, they have double contact along 
the polar. 

271. Tftivo conies have double contact, and through the points 
of contact a third conic is drawn, its corresponding common chords 
with each of the conies will intersect on the chord of contact. 
Art. 215. 



246 



CROSS-RATIO GEOMETRY 



[CH. XVII 



272. 7/ two conies A and B have each double contact with a 
third conic C, a pair of their common chords pass through the 
point of intersection of the two chords of contact^ and form a 
harmonic pencil with them,*. 

For the intersection P of the chords of contact is the pole of 
the line joining the poles of contact T, T\ and therefore has the 
same polar for all three conies; consequently, by Art. 212, it is 
the intersection of a pair of common chords of A and B. 

PT, PT' form with the pair of common chords through P a 
harmonic pencil by Art. 230. 

273. If any transversal through the pole of cmitact T meets 
the conies in aa\ bb', then as the transversal rotates about T the 
value of the cross-ratio {Tbaa') remains constant, and is the re- 
ciprocal of (Tb'aa). 




Fig. 119. 

Let Ta^b-fi-^a^ be any other position of the transversal. 

Then, by Art. 244, aaj, bb^ meet on the chord of contact, as 

do also «%, a'oi', by Art. 161. Therefore the three chords 

* Poncelet, Prop. Prqj. Art. 427; Chasles, Sect. Con. Art. 415; Salmon, 
Art. 263. 



272-274] coNics having double contact 



247 



I 



fiai, aa^j bb^ all pass through the same point G on the chord 
of contact, and by Art. 161 b'bi passes through the same point. 
Therefore the ranges {Tbaa') and (Tb^a^a^) have T for a corre- 
sponding common point, and are in perspective, centre G^ and 
their cross-ratios are consequently equal by Art. 21. And since 
these are any two positions of the transversal, the cross-ratio 
{Tbaa) is the same for all positions of it. 

Again, by Art. 230 T is one of the double points of the 
involution determined by (aa', bb'). 



Therefore {Tbaa') = {Tb'a'a) 
Conversely we have : 



1 



{Tb'aa) 



by Art. 3. 



If through a given point T a transversal is drawn meeting 
a conic in aa, and on it a point b is taken such that the cross-ratio 
{Tbaa') is constant, the locus of b is a conic having double contact 
with the given conic along the polar of T. 

274. The chords which join pairs of corresponding points of 
two homographic rotes on a conic envelop a second conic which has 
double conta>ct with the given one at the common points of the rows, 

M' H 




Fig. 120. 



248 CROSS-RATIO GEOMETRY [CH. XVII 

Let AA\ BB' be pairs of corresponding points of two home 
graphic rows on the conic. Let jE', F be the common points of 
the rows. Then by Art. 157 EF is the cross-axis on which paii 
of chords joining pairs of corresponding points taken inverselyl 
intersect. Let AB\ A'B intersect on EF in M, and produce] 
AA\ BB'... to meet the tangents at E, F in aa\ hh' .... We will" 
shew that these are pairs of corresponding points of two homo- 
graphic ranges. 

Consider the inscribed quadrangle ABA'B'. Its diagonal 
points are G^ H, M. Therefore HG is the polar of if for the 
conic. But since M is on EF, its polar passes through T. There-, 
fore TGH is a straight line, and the following are harmonic : 
G{TMAB), (Tmab), {Tm'a'b'), (Tm'bW). 

Therefore (Tmab) = (Tm'b'a'), and ab', ba intersect on mm, 
i.e. on MG. 

Let them intersect in M'. Join TM'. 

Then in the quadrangle aba'b', T, G, M' are the diagonal 
points. Therefore TG is the fourth harmonic of TM' for TE, TF. 
But from the conic, since TG is the polar of i/, TG is the fourth 
harmonic of TM for the tangents TE, TF. Hence TM' coincides 
with TM, and M' with M. Therefore ab', a'b intersect in M, 
i.e. on the fixed line EF, which is consequently the cross-axis of 
the two homographic ranges of which aa', bb'... are pairs of 
corresponding points. Therefore, by Art. 139, aa', bb' ... are 
tangents to a conic which touches TE^ TF at the points E, F. 
Conversely, 

If two conies C and C' have double contact at E, F, and a 
chord A A' of C rolls upon C, its extremities A, A' form two homo- 
graphic conic-j)encils whose common points are at E, F. 

Cor. 1 . If TP, TQ are two fixed tangents to a conic, R and 
S two variable points on the curve such that either (1) (PQRS) or 
(2) T(PQBS) is constant, the chord PS will envelop a conic having 
double contact with the given conic at P, Q, and conversely/. 



I 



274-275] 



ENVELOPE THEOREM 



249 



For by Arts. 159, 192 {R) and {S) form two homographic 
divisions on the conic, having P, Q for common points. 

Cor. 2. If tangents are drawn at pairs of corresponding points 
of two homographic divisions on a conic, the locus of their point 
of intersection is a conic having double contact with the former at 
the common points of the rows. 

This theorem, which is the correlative of Art. 274, can easily 
be proved by the method of Art. 275. 

275. If the locus oj a point a is a conic C, the enveloj^e of its 
polar for a conic C is a conic C" ; and conversely, if a straight 
line OKi' moves so as to envelop a conic C'\ the locus of its pole for 
a conic C is a conic C*. 




Fig. 121. 

Let P, P' be two fixed points on C, and let a, fi, y... be any 
other points on C. Let OA, OA' be the polars of P, P' for C", 
and let the polars of a, ft, y... for C" meet OA in a, h, c... and 
OA' in a', h\ c' .... 



Poncelet, Fro^. Proj. Art. 231. 



250 CROSS-RATIO GEOMETRY [CH. XVII 

Then Pa is the polar of a, and P'a the polar of a\ &c., and] 
since by Art. 167 

the range of poles {ahc. ..) = the pencil of polars P (aySy . . . ), 
and 

the range of poles (a'6'c'...) = the pencil of polars P' (a^y...), 
and by Art. 129 P {a^y . . .) = F {a^y . . .), 
therefore {ahc . . . ) = {a'h'c' . . . ), 

and by Art. 139 the lines aa, hh' ... envelop a conic touching OAi 
and OA'. 

To prove the converse, let OA^ OA' be two positions of the- 
moving line aa', and let P, P' be the poles of OA^ OA' for C 

Then by Art. 1 30 {abc. ..) = {a'h'c . . . ), 

therefore by Art. 167 P (a/8y . . . ) = P' (a^y . . . ), 
and by Art. 138 the locus of a is a conic through P and P' . 

276. If a triangle ABO inscribed in a conic moves so that' 
two of its sides pass through fixed points P, P', its third side wilV 
envelop a conic having double contact with the given conic at the 
points where the latter is met by the line PP'*. 

As the triangle moves, the conic pencils {A) and {B) being 
homographic to (C) are homographic to each other by Art. 
186. Therefore by Art. 274 AB envelops a conic having 
double contact with the given conic at e and f. 

277. If a triangle circumscribing a conic moves so that its 
base angles move along fixed straight lines OM, OM', its vertex 
will describe a conic having double contact with the given conic 
at the points where the latter is met by the polar of Of. 

In Fig. 122 through A, B, draw the tangents forming the 
circumscribing triangle abc, and let be the pole of PP'. Then 

* Poncelet, Prop. Proj. Art. 431. 

+ Poncelet, Art. 435 ; Salmon, Art. 272, Exs. 2, 3. 



276-277] coNics having double contact 251 

fO 




Fig. 122. 



252 CROSS-RATIO GEOMETRY [CH. XVII 

as the triangle ABC moves it is evident that the points a, 6, 
being the poles of BC, CA, will move along the fixed lines 
OM, OM' the polars of P, P', and by Art. 275 the point c will 
describe a conic. Then use Art. 274, Cor. 2. 



EXAMPLES. 

1. A, B are two fixed points on a conic, and L is a given straight line 
meeting it in M, N. P is a variable point on L, and AP, BQ meet the conic 
again in Q, R. Shew that the envelope of QR is a conic having double 
contact with the given conic at the points M, N, and touching the line AB. 

2. CA, GB are two given tangents to a conic, and D a fixed point in the 
plane. Any transversal through D meets CA, GB at E, F, and from E, F 
other tangents are drawn meeting in T. Shew that the locus of T is a conic 
passing through C, and having double contact with the given conic at the 
points of contact of the tangents from D. 

3. Two conies A, B have double contact at Q, E; T being the pole. 

(1) If P is a variable point on A, and PQ, PR meet B in D, J5J, then 
BE envelops a conic having double contact with A and B at Q and R. 

(2) If FG is any chord of A which is also a tangent to B, and if QF, RG 
intersect in H, the locus of If is a conic having double contact with A and 
P at Q and R. 

4. If two conies have double contact the cross-ratio of four of the points 
in which any four tangents to the one meet the other is the same as that of 
the other four points in which the four tangents meet the curve, and also 
the same as that of the four points of contact. [Townsend.] 

5. aa', bb', cc' are three fixed chords of a conic. Shew that the envelope 
of a fourth chord dd' such that {abcd) = {a'b'c'd') is a conic having double 
contact with the given conic. 

6. The locus of the intersection of tangents to a conic which divide a 
finite segment II' of a given tangent in a constant cross-ratio is a conic 
having double contact with the given conic at the points of contact of tangents 
from I and I\ 




EXAMPLES 258 

7. Two conies A and B have double contact &t Q, R. If through two 
points m, m' on B we draw tangents to it meeting A in the two pairs of points 
a, b and a', b', then the two chords aa\ bb' will pass through the intersection 
of the lines QR and mm'. 

8. In Ex. 7 if from two points n, n' on A we draw chords touching B 
and forming the circumscribed quadrilateral abed, one diagonal of this 
quadrilateral will pass through T the pole of contact of the conies, and 
through the pole of nn' for A. 

9. In Ex. 7, if the vertex p of an angle circumscribing B moves along A, 
the points m, m' where the sides of the angle meet A form two homographic 
divisions which have Q, R for common points, and the chord mm' envelops 
a conic having double contact with the given conies at Q, jR. 



CHAPTER XVIII 

CONSTRUCTION OF A CONIC SATISFYING CERTAIN 
CONDITIONS 

278. To describe a conic through jive given points. 

This problem has been fully solved in Art. 140 by what we 
may term the first method, due to Chasles. On account of the 
importance of the question, and the frequent reference that is 
made to it in constructions connected with the conic, we have 
given a few other methods so that when a student is told to 
"describe a conic through five points" he may select any one 
of the methods and know exactly what the words imply. 

Second method. 

Let a, 6, c, 0, 0' be the given points. By Art. 138 the locus 
of the intersections of corresponding rays of two homographic 
pencils not in perspective is a conic passing through the centres 
of the pencils. If then we take two of the five points 0, 0' as 
centres, and join each of them to the remaining three points a, 6, c 
we shall obtain two pencils each containing three rays, and if 
through one of the centres, 0, we draw any fourth ray Od, and 
construct the ray corresponding to it in the second pencil, the 
point 8 where these two rays intersect will be a point on the 
conic. This problem is solved completely in Art. 48, and by 
drawing different rays through 0, and repeating the construction, 
we can obtain as many points 8 on the curve as we please. 

Third m>ethod, employing Pascal's Theorem, Art. 146. 

Through y the intersection of 06, O'a draw any straight line 



278-279] 



CONSTRUCTIONS OF CONICS 
b 



255 




Fig. 123. 

meeting O'c in a and Oc in ft. Join a^ and 6a meeting in c. 
Then by Art. 147 c' is a point on the curve, and by drawing 
different lines through y, and treating them as Pascal lines, 
we can obtain other points on the curve. 

Fourth method, employing Maclaurin's Theorem, Art. 151. 

In Fig. 123 let aO\ Ob meet in y. Through a draw any 
straight line ac meeting Oc in fi, and join ^y meeting O'c in a. 
Then g\ the intersection of a^ and ba, is a point on the curve, 
for it is the vertex of the triangle cap whose base angles /8 and a 
move along the fixed lines cO, cO\ and whose sides pass through 
the fixed points a, 6, y. 

Fifth method, by Desargues' Theorem, Art. 187. 

Consider OacO' as an inscribed quadrangle, and through b 
draw any transversal meeting the opposite sides ac, 00' in d, d\ 
and the diagonals O'a, Oc in e, e', and on the transversal find the 
point c' such that {dd\ ee\ be) is an involution range. Arts. 104, 
\0%adjin. Then by Art. 187 c is a point on the required conic. 

Other methods might be given, but these are sufiScient to 
shew the application of the theory of cross-ratio to the problem. 

279. To draw the tangent at 0\ any one of the fine given 



256 



CROSS-RATIO GEOMETRY 



[CH. XVIII 



If we take a point d on the curve very near to 0', since Od, O'd 
are corresponding rays in the two homographic pencils centres 
and 0\ it is obvious that the tangent at 0' is the ray in the 
pencil whose centre is 0' corresponding to the ray 00' in the 
pencil centre ; and consequently the rays constructed in 
Art. 58 are the tangents at and 0', their intersection T being 
the cross-centre of the pencils. 

280. To find the points where the conic through five points 
meets a given straight line L. 

Let the pencils 0{abc) and 0' (abc) meet L in the points 
a, 13, y and a, yS', y. Then the points required are obviously the 
common points of the two homographic co-axial ranges of which 
aySy and a'jS'y' are the characteristics. This problem is solved in 
Arts. 83—86. 

281. To find the directions of the asymptotes oj the conic 
through five points. 




Fig. 124. 



280-284] CONSTRUCTIONS OF CONICS 257 

This is equivalent to the problem of finding a pair of parallel 
corresponding rays in the two pencils centres 0, 0'. 

Through draw three rays Oa\ Ofi', Oy parallel respectively 
to O'a, 0'6, O'c. Then we have two concentric pencils whose 
characteristics are {abc) and (a')8 y'), and if we cut them by 
any transversal Z, the common rays Oe, Of, which can be found 
bj' Art. 84, give us the required directions. 

282. To draw the asymptotes of the conic through five points. 
In Fig. 124, -through a, h, c draw pairs of lines respectively 

parallel to Oe and Of (Art. 281), meeting any transversal L' in 
a-Jy^c^ and a-^h^c^. The parallel lines aa^, bb^, cc^ are three rays 
of a pencil whose centre is at infinity along Oe, and aa^^, bb^, cc( 
are three rays of a pencil whose centre is at infinity along Of 
Consider the two ranges whose characteristics are a^^c^ and 
a^b-^c^. Find by Art. 82 / the point on the first range corre- 
sponding to the point at infinity on the second, and J' the point 
on the second corresponding to the point at infinity on the first. 
Then the line through / parallel to Oe is one asymptote, and the 
line through J' parallel to O/is the other. 

283. To construct a conic given five tangents. 

In Fig. 79 let the five tangents be ab, aa, a'b', bb', cc . 

Let ab, a'b' meet in 0. Consider the ranges on Oa, Oa' whose 
characteristics are abc and a!b'd . Find d, d' any pair of corre- 
sponding points by Art. 40. Then by Art. 139 dd' is a tangent. 

Similarly any number of tangents can be drawn, and the 
conic constructed by means of them. 

284. Given five tangents, to find the point of contact of any 
one of them, a'b' suppose. 

First method. 

Find by Art. 40 the point A' in the range a'b'c' corresponding 

M. 17 



258 CROSS-RATIO GEOMETRY [CH. XVIII 

to in the range ahc. By Art. 130 ac^ fin. A' is the point 
required. 

Second method. 

In Fig. 79 join ac' meeting a'c in 8. Then 68 will meet a'b' 
in the point required. For if d' moves up to and coincides with 
A', y will coincide with c', etc. 

285. Given five tangents, to draw a pair of tangents from a 
given point P. 

In Fig. 79 let the given tangents be ah, aa\ a'h\ hh\ cc. 
Join P to the points a, 6, c and a', h\ c'. Then the required 
tangents are evidently the common rays of the pencils whose 
characteristics are P{abG) and P{a'b'c'), and can be constructed 
as in Art. 84. 

286. To construct a conic given four points and a tangent. 

Let a, b, c, d be the four points, L the given tangent. 
Consider Z as a transversal meeting the opposite sides of the 
quadrangle abed in the pairs of points aa', ySyS'. Then by Des- 
argues' Theorem, Art. 187, e,y the double points of the involution 
determined by (aa, fi/3') are the points of contact of L with the 
two conies which satisfy the conditions of the problem. These 
can then be constructed by one of the methods of Art. 278. 

287. To construct a conic given four tangents and a point. 
Let P be the given point, and join P to the pairs of opposite 

vertices of the quadrilateral formed by the four given tangents. 
Then by Art. 188 these four rays determine a pencil in involu- 
tion in which the double rays are tangents to the two conies, 
which can then be constructed from the two sets of five tangents 
by Art. 283. 

288. If a system of conies is described passing through two 
given points a, 6, and touching two given straight lines OT, OT', 
(I) the polar 8 of the point pass through one or other of two fixed 



H 285-288] 
^V points on t 



CONSTRUCTIONS OF CONICS 



269 



points on the line ab^ and (2) the poles of the line ah lie on one 
or other of two fixed lines passing through 0. 




Fig. 125. 

Let d, e be the points where any conic of the system touches 
OT, 0T\ so that de is the polar of for that conic. Then the 
segment de may be considered as a quadrilateral inscribed in the 
conic, two of its opposite sides being the tangents at d and e, 
the other pair of opposite sides being represented by de con- 
sidered as two coincident straight lines. 

(1) Let ah meet OT, OT in the points T, T', and de in /. 
Then by Desargues' Theorem, Art. 187, the two pairs of points 
(a6), {TT') determine a system in involution in which / is one 
of the double points; and if/' is the harmonic conjugate of/ 
for a and 6, /' is the other double point of the involution, 
by Art. 99. Hence the polars of for the different conies of 

17—2 



260 CROSS-RATIO GEOMETRY [CH. XVIII 

the system pass through one or other of the known points 

(2) Again, considering the same conic we see that Q/*, Of 
are harmonic conjugates both for (Oa, Oh) and for {OT^ OT'), 
and therefore by Art. 161 the pole of ah lies on one of the lines 
Of or Of. Hence the poles of ah for the dififerent conies of the 
system lie on one or other of the known lines 0/*, Of. 

289. To construct a conic given three points and two tangents. 

Let a, 6, c be the three points, OT, OT' the two tangents. 

It was shewn in Art. 288 that for the system of conies 
passing through the points a, h and touching the lines OT^ 0T\ 
the polars of the point pass through one or other of two known 
points /, /' on the line a6, see Fig. 1 25. Similarly for the system 
of conies passing through the points 6, c and touching the same 
pair of lines OT, 0T\ the polars of the point pass through one 
or other of two known points g, g on the line he. 

Now the conies which pass through the three points a, h, c, 
and touch the lines OT, 01", are those which are common to the 
above two systems, and are therefore such that the polars of O 
pass through one of the points f,f, and also through one of the 
points g, g'. Hence there are four, and only four, polars, and 
consequently four, and only four, conies. If one of the polars 
meets OT, OT' in d, e, the problem is reduced to the construction 
of a conic through five points, Art. 278. 

290. To construct a conic given three tangents and two 
points. 

Let the points be a, h and the tangents OT, OT', TT', and 
let the line ah meet them in the points t, t', t" respectively. 

It was shewn in Art. 288 that for the system of conies 
passing through a, h and touching the lines OT, OT' the poles 
of the line ah lie on one or other of two known lines passing 
through 0, viz. Of, Of, where/,/' are the double points of the 




289-290] CONSTRUCTIONS OF CONICS 261 

involution determined by (a6, tt'). Similarly for the system of 
conies passing through a, h and touching the lines OT^ TT', the 
poles of the line ah lie on one or other of two known lines 
passing through T, viz. Tf^, Tf-[^ where /j,// are the double points 
of the involution determined by (a6, tt"). 

Now the conies which pass through a, h and touch the three 
straight lines OT^ OT', TT' are those which are common to the 
above two systems, and are therefore such that the poles of ah 
lie on one of the lines 0/*, Of\ and also on one of the lines 
Tf^f Tf^. Therefore there are four, and only four, poles, and 
consequently four, and only four, conies. 

If one of the poles is P, then Pa and Fh are tangents, and 
the problem is reduced to the construction of a conic touching 
five lines, Art. 283. 



CHAPTER XIX 

HOMOGRAPHIC GENERALISATION OF CIRCLES AND THE 

CULAR POINTS AT INFINITY, CONICS AND THEIR FOCI, 
AND OTHER ASSOCIATED POINTS AND LINES 

291. We touched briefly on the relations of the circular points 
with points and lines in Arts. 113, 114, and with circles and conies 
in Arts. 179 — 183. We will now consider them more fully. We 
shall use small letters to denote points in the original, and their 
capitals to denote the corresponding points in the generalised 
or derived figure. As the equations to the isotropic lines joining 
the origin to the circular points are y = ±ix, we shall always 
denote these points in the original figure by the letters i, i', and 
their generalised positions by their capitals, /, /'. Since every 
circle passes through i, i' , the points /, /' will only lie on a conic 
when it is generalised from a circle in the original figure. If 
a conic generalises into another conic so that the focus of the 
first becomes the intersection of two tangents to the second, 
the points /, /' are finite points on these tangents but not on 
the curve, being the points of contact only when the original 
conic is a circle, its centre, the pole of the line at infinity, 
becoming the intersection of the tangents, i.e, the pole of //'. 

We will first give a list of the more important fundamental 
results which are obtained from the consideration of the circulai 
points. These the student will easily verify from what we havt 
said on the subject in the above quoted articles, and we will' 
then shew how these results can be applied to obtain generalised 
properties of conies from the known properties of circles. 



DATA 1-10] HOMOGRAPHIC GENERALISATIONS 263 

Data. 

1. The line at infinity becomes a finite line, on which are 
two finite points /, /' corresponding to i, i' the circular points at 
infinity. 

2. If c is the mid-point of a linear segment ab, and d the 
point at infinity on the line, and ii A,B,C,D are the generalised 
positions of a, 6, c, c?, then D lies on //', and (A BCD) is harmonic. 

3. If c divides ab in a given ratio, with the notation of 2, 

/■in 

(ABCD) = (aic^) = ^^. 

4. Lines which are parallel in the original figure become 
lines intersecting on //'. 

5. Pairs of concurrent lines at right angles become pairs of 
lines which cut the segment //' harmonically. 

6. Pairs of concurrent lines containing a constant angle 
become pairs of lines which cut the segment //'in a constant 
cross-ratio. 

7. Pairs of concurrent lines containing angles bisected by 
a single pair of lines become pairs of concurrent lines cutting 
the segment IT in a series of points in involution in which /, /' 
are conjugate points, and the bisectors cut //' in the double points 
of the involution. 

8. A circle becomes a conic through the points /, /', and the 
centre of the circle becomes the pole of //'. 

9. A figure consisting of a conic, a pole and its polar can 
represent a circle, its centre, and the line at infinity. 

10. A circle on ab as diameter becomes a conic through 
A, B, /, /', and having AB and //' for a pair of conjugate 
chords. 

Since only one circle can be described on a given finite 
straight line as diameter, it follows that if we have given two 



I 



264 CROSS-RATIO GEOMETRY [CH. X 

pairs of points in a plane, only one conic can be described passing 
through them, and having the line joining one pair conjugate to 
the line joining the other pair. 

11. A focus is equivalent to the intersection of two tangents 
passing through the points /, I' which are not on the conic. 
The other focus becomes the intersection of the other tangents 
from /, /'. 

12. A straight line through a given focus becomes a straight 
line through the intersection of two given tangents. 

13. The tangents from the foci intersect in i, i\ and 
therefore to have given two foci is equivalent to having given a 
quadrilateral circumscribing a conic, /, /' being a pair of opposite 
vertices. 

14. Confocals become conies inscribed in a quadrilateral 
having /, /' for a pair of opposite vertices. 

15. A parabola touches the line at infinity, and S being the 
focus, it has aS'i, Si' for tangents, and therefore Sii' is a tangent 
triangle ; hence a parabola and its focus become a conic inscribed 
in a given triangle. 

16. A rectangular hyperbola, having its asymptotes at right 
angles, has i, i' for conjugate points, and therefore becomes a 
conic cutting the segment //' harmonically. 

17. Concentric circles become conies having double contact 
with one another at /, /'. 

18. Conies having the same focus and directrix become conies 
having double contact. 

19. Similar conies, having the angles between their asym- 
ptotes constant, become conies cutting the segment //' in a 
constant cross-ratio. 

20. Co-axial circles become conies circumscribing the same 
quadrangle, two of whose vertices are /, /', the other two 



DATA 11-22] HOMOGRAPHIC GENERALISATIONS 265 

vertices being the points A, B^ corresponding to the points 
where the radical axis meets the circles. 

In Fig. 102 let g^ = A, g^ = B, g, = I, g,= /'. 

The line of centres becomes the line containing the poles 
of //', and is therefore the line P^P^, one of the sides of the 
common self -conjugate triangle, and p is the intersection of the 
radical axis and the line of centres. Since these two lines 
are at right angles, {g^g^j)'P^ is harmonic by 5 supra. 

Limiting points. This term is a little misleading. Perhaps 
it would be better to call them limiting circles, as they are 
limiting forms of circles of the system. There are three limiting 
circles, two of them being point circles on the line of centres, and 
the third a circle of infinitely large radius consisting of the radical 
axis and the line at infinity. 

The limiting point circles when considered as points become 
Pzf Psi two of the vertices of the common self -conjugate triangle. 
When considered as circles they coincide with their asymptotes 
by Art. 179, and become respectively the pairs of common chords 
through Pg and Pg. 

The limiting circle of infinitely large radius becomes the pair 
of common chords through the third vertex P^. 

The property that the radical axis bisects the segment joining 
the limiting point circles becomes (ppP^Ps) = — 1. 

Of any pair of common chords one can be taken to represent 
the radical axis, and the other to represent the line joining the 
circular points in the original figure. 

21. The centres of similitude of two circles become a pair of 
tangent vertices. 

22. The condition that two chords pq, p'q of a circle, centre 
t, are equal, is equivalent to either of the conditions 

T(irPQ)=T(irPQ'), 

or the c.p. (IPPQ) = (IP P'Q'). 



266 CROSS-RATIO GEOMETRY [CH. XIX 

23. Two orthogonal circles, centres c, t, intersecting in a, 6, 
possess the following fundamental properties, from any one of 
which the others can be deduced : 

(1) The tangents at a point of intersection are at right angles. 

(2) The tangents at a point of intersection pass through the 

centres. 

(3) The centre of one circle is the pole of the common chord 

for the other circle and the pole of W for its own 
circle. 

In (1) and (2) it follows that if the property is true for one 
point of intersection, and in (3) for one centre, it is true for the 
other also, and any one of the three properties might be taken 
as defining two orthogonal circles. 

Consequently, if we generalise two orthogonal circles we 
shall obtain two conies a, ^ which will possess the following 
properties : 

Let //', A B he Si pair of common chords, T, C the poles of 
//' for a, ft see Fig. 100. Then 

(1) The tangents at A divide //' harmonically. 

(2) The tangents at A pass through T^ C. 

(3) T is the pole of AB for ft 

In (1) and (2) it follows that if the property is ti'ue for the 
point A corresponding properties hold for the tangents at B^ and 
in (3) C is the pole of ^^ for a. From the property in (3) for 
convenience of reference we have called two conies which are so 
related contra-polar conies, or from the property in (1) they 
might be called harmotomic conies as explained in Arts. 203, 
204. Hence a pair of orthogonal circles become a pair of contra- 
polar conies. 



I 



DATUM 23] 



EXAMPLES 



EXAMPLES. 



267 



In the column on the left are given the elements of the original figure 
which the student should draw for himself. In the column on the right will be 
found the corresponding elements and properties of the generalised theorems. 

In every case the letters I, I' are the representatives of the circular 
points at infinity, and when I, I' are on the curve, T the pole of II' repre- 
sents the centre of the circle. 

The letters have been so chosen that Fig. 126 will apply to Examples 
1—14. 

L 




Fig. 126. 



268 



CROSS-RATIO GEOMETRY 



[CH. XIX 



1. The angle contained in the The cross-ratio of four fixed points 

same segment of a circle is constant. on a conic is constant. Art. 129. 

Let a, & be two fixed points on a circle, p a variable point on it, and let 
pa, pb meet the line at infinity in a, /3. Then by 6 supra p (a^ii') is 
constant. If the circle becomes a conic we have P (^BII') = constant. 



2. The tangent at any point of 
a circle is at right angles to the 
radius through the point of contact. 



Any chord of a conic is cut har- 
monically by any tangent and the 
line joining its point of contact to 
the pole of the chord. Art. 178. 
The line at infinity becomes a chord cutting the conic in I, I'. The 
centre of the circle becomes T, the pole of II'. The radius of the circle 
becomes the line joining I' to any point A on the curve. Then by 5 supra 
TA and the tangent at A cut II' harmonically. From this we can at once 
deduce that if a variable tangent meets two fixed tangents, it is divided 
harmonically by them, their chord of contact and the curve. Also if the 
tangent at A meets II' in JT, TA and TK are conjugate lines since they 
divide II' harmonically. 



3. Any diameter of a circle is 
bisected at the centre. 



Any chord through a given point 
is divided harmonically by the curve, 
the point, and its polar. Art. 161. 



4. If pq is a diameter of a circle If IF is a chord of a conic, T its 

centre t, the c.p. {pqii') is harmonic. pole, and TPQ a chord through T, 

the c.p. {PQII') is harmonic, i.e. 
PQ, II' are conjugate lines. Art. 171. 
Two chords are conjugate if either passes through the pole of the other. 
Art. 165. 



5. The angle in a semi-circle is 
a right angle. 



In Ex. 4, i? is any point on the 
curve. If PR, QR meet II' in D, E, 
then {II'DE) is harmonic. Art. 
208 (7). 



6. If a straight line through the 
centre of a circle bisects a chord 
which does not pass through the 
centre, it cuts it at right angles ; and 
conversely, if it cuts it at right angles, 
it bisects it. 



Given a chord of a conic II', T 
its pole, and any chord PR cutting 
II' in D, and S on PR so that 
(PRDS) is harmonic. Let TS meet 
II' in E. Then {II'DE) is harmonic; 
and conversely, if {II'DE) is har- 
monic, so also is {PRDS). 
Also, since T{irDE) and T(PRDE) are harmonic, T {PR, II', DE) 
is an involution pencil, of which TD, TE are the double rays. Art. 208 (5). 



EXS. 1-11] 



EXAMPLES 



269 



7. ca, cb are tangents to a circle, 
centre t ; ab and ct meeting in g. 
(1) ct bisects the angle atb. 



(2) ct bisects the angle acb. 



(3) ct bisects ab at right angles 



atgr. 



II', A B are two chords of a conic, 
T, C their poles. 

(1) TC is one of the double rays 
of the involution pencil T {II\ AB). 
Art. 208 (f). 

(2) If CT, AB, AG, BG meet 
ir in F, H, K, L, GT is a double 
ray of the involution pencil 

G [II ', KL). 

(3) The ranges (ABGH) and 
{II' FH) are harmonic. 



Hence TG and TH are the double rays of the involution pencil 
T{ir, AB, KL). 

From the above we also obtain the properties : 

Tangents to a conic subtend equal angles at the focus, and if T is the 
focus and G the pole of a chord AB, then AB is divided harmonically by 
GT and the directrix, the polar of T. 

8. If the tangent at any point Given a chord II' and its pole T, 

a of a conic meets the S directrix if the tangent at any point A meets 
in k, aSk is a right angle. II' in K, TA and TK are conjugate 

lines. See Ex. 2 ad fin. 



9. ab is a fixed chord of a circle, 
c its pole, and r any point on the 
circumference. The bisectors of the 
angle arb pass through two fixed 
points, viz. the extremities of the 
diameter passing through c. 

10. Two parallel tangents to a 
circle intercept on any variable 
tangent a segment which subtends 
a right angle at the centre. 



11. If ca, cb are tangents to a 
circle, centre t, the circle round abc 
has ct for a diameter. 



II', AB are two given chords of a 
conic, T, G their poles. GT meets 
the conic in N, 0. If R is any 
variable point on the curve, the 
double rays of the involution pencil 
R {AB, II') always pass through the 
points N, 0. Art. 164 ad fin, 

II' is a given chord of a conic, T 
its pole. K is & point on II'. KA, 
KA' are two tangents cut at C, C" by 
the variable tangent at B. Then 
T{II'GG') is harmonic. Art. 175 
(1). 

AB, II' are two chords of a 
conic, G, T their poles. The six 
points mentioned lie on a conic and 
GT is the polar of the intersection of 
AB and II' for both conies. Art. 199. 



270 



CROSS- RATIO GEOMETRY 



[CH. XIX 



12. Chords of a circle which 
subtend equal angles 

(1) at the centre, 

(2) at the circumference, 
envelop a concentric circle. 



II' is a given chord of a conic, 
T its pole, R, Q two points on the 
curve such that 

(1) T{II'RQ) is constant, 

(2) the c.p. {II'RQ) is constant, 
RQ envelops a conic having double 
contact with the given conic at I, I'. 
Arts. 159, 192, 274. 

13. The envelope of the chord of a conic which subtends a constant 
angle at the focus is a conic having the same focus and directrix ; and so is 
the locus of its pole. 

The focus and directrix become a pole T and its polar UV. Let I be on 
TU and 2' on TV, Then corresponding to the moving chord pq of the given 
conic in the original figure we have a chord PQ in the generalised figure such 
that T (UVPQ) is constant, and as in Ex. 12 (1) PQ envelops a conic having 
double contact with the other, and similarly for the locus of the pole of PQ. 

14. In any conic the intercept on a variable tangent made by two 
fixed tangents subtends a constant angle at the focus. 

Here I, I' are points (other than the points of contact) on the tangents 
from T. In Fig. 126 let a variable tangent meet the tangents from T'mti, 1-2, 
and those from G in Ci, C2. Then by Data No. 6 the generalised property 
becomes: T {II'ciC2) is const, i.e. (^1^20102) is const, and we have the anhar- 
monic property of tangents, Art. 130. 




Fig. 127. 



Exs. 12-19] 



EXAMPLES 



271 



[The letters in Examples 15—19 refer to Fig. 127.] 



15. If ap is any chord of a circle, 
pq the diameter through p, and p7n 
the perpendicular on the tangent at 
a meeting the circle in p', then ap 
bisects the angle p'pq, and p'q is 
parallel to the tangent at a. 



II' is a given chord of a conic, T 
its pole, TPQ a chord through T. 
The tangent at any point A meets 
II' in K. ilf is the harmonic conju- 
gate of K on ir, PM meets the 
curve in P'. Then PA is a double 
line of the involution pencil P (II'QP') 
and QP' passes through K. 



16. The envelope of a chord uv 
of a circle which subtends a right 
angle at a fixed point c not on the 
curve is a conic having the fixed 
point and the centre of the circle 
for foci. 



17. c is a fixed point in the 
plane of a circle, u any point on the 
curve. If vd is drawn making a 
right (or any constant) angle with 
uc, ud envelops a conic having c for 
focus. 



18. c is a fixed point in the 
plane of a circle, centre t. The 
locus of the mid-points of all chords 
through c is the circle on ct as 
diameter. 



19. The locus of the points 
where parallel chords of a circle are 
cut in a given ratio is an ellipse 
having double contact with the circle 
at the extremities of the diameter 
perpendicular to the chords. 



ir is a fixed chord of a conic, T 
its pole, and C a point not on the 
curve. If UV is a variable chord 
such that C(II'UV) is harmonic, 
UV envelops a conic inscribed in the 
quadrilateral CITI', and CT is a 
side of the common self -conjugate 
triangle of the two conies. 

II' is a given chord of a conic, 
C a fixed point in its plane. Any 
straight line through G cuts the 
conic in U, and II' in E, and 
{II'ED) is harmonic (or constant). 
Then UDW envelops a conic touch- 
ing CI and CI'. 

II' is a chord of a conic, T its 
pole, C a fixed point in its plane. 
Any chord UU' through C meets II' 
in E and {UU'EX) is harmonic. 
The locus of X is a conic through 
C, T, I, I' and having CT, W con- 
jugate chords. Art. 191. 

If through a fixed point C a 
straight line is drawn meeting a 
conic in C7, U', and on it a point Y 
is taken such that (UU'CY) is con- 
stant, the locus of F is a conic 
having double contact with the given 
conic at the extremities of the polar 
of C. Art. 273. 



272 CROSS-RATIO GEOMETRY [CH. XIX 

20. Two pairs of the lines join- II' is a chord of a conic, T its 
ing the extremities of two diameters pole, TAB, TA'B' are two chords 
of a circle are parallel, and the other through T. Then if ^^', BE' meet 
two pairs are at right angles. in C, and AB', A'B meet in D, the 

points 0, D will lie on II' and divide 
it harmonically. 

21. The locus of the mid-points If a series of chords of a conic all 
of a series of parallel chords of a pass through a point D, the locus of 
circle is the diameter perpendicular the harmonic conjugates of D on the 
to the chords. chords is a straight line on which 

are situated the poles of all the 
chords. Art. 161. 
Let AB be any chord through D, C its pole, and R a point on it such 
that {ABBB) is harmonic. Let T be the pole of any chord PQ not passing 
through D, and let CR cut PQ in E. Then E lies on the polars of D and T. 
Hence any chord through E, such as PQ, is cut harmonically at E and the 
point where it meets DT. Therefore TE, TD are harmonic conjugates of 
the tangents TP, TQ. Now draw any transversal through D, cutting TP, 
TQ in the points I, I', and transform so that I, I' become i, i' . Then D is 
on the line at infinity, and ORE becomes the locus of the mid-points of a 
system of parallel chords of which TB is the direction. Moreover, T is now 
a focus, and PQ the corresponding directrix, and TD, TE .being conjugate 
for Ti, Ti', are at right angles to each other. Hence we have the theorem: 
' * The locus of the mid-points of a series of parallel chords of a conic is a 
straight line which cuts a directrix in a point E such that the corresponding 
focal distance TE is perpendicular to the system of chords. Also the poles 
of the system of chords all lie on the locus of their mid-points." 

22. If a line is drawn through T^ , TjB are two given tangents to 
a focus of a central conic making a a conic, I, I' given points on them, 
constant angle with a tangent, the If a variable tangent at meets 
locus of their intersection is a circle. TA, TB, II' in P, Q, R respectively, 

and (PQRS) is constant, the locus of 
S is a, conic through I, I'. 

[In the original figure let s be the focus, and let st meet the tangent 
at p in t so that stp is constant, and draw sy perpendicular to pt. Then the 
triangle syt is of constant species, and since y describes a circle, the locus of 
f is a circle.] 

If the line II' is at an infinite distance so that R is at infinity, and PQ 
is divided in a constant ratio at S, the theorem becomes "The locus of the 
point where the intercept on a variable tangent made by two fixed tangents 
is divided in a constant ratio is a hyperbola whose asymptotes are parallel 
to the given tangents." Chap. XI, Ex. 13. 



Exs. 20-27] 



EXAMPLES 



273 



If the original conic is a parabola, the locus, instead of being a circle, is 
a straight line; and as II' is now a tangent, the property becomes " If a 
variable tangent to a conic meets three fixed tangents in P, Q, R, and 
{PQRS) is constant, the locus of S is a straight line." If one of the 
tangents, as II', is at infinity, the generalised conic is a parabola, one of 
the three points, as R, is at infinity, and the theorem becomes "The locus 
of the point where the intercept on a variable tangent to a parabola made by 
two fixed tangents is divided in a constant ratio is a straight line." See 
Chap. VI, Ex. 26. 



23. The locus of the intersection 
of tangents to a conic at right angles 
is a circle. 



The locus of the intersection of 

tangents to a conic which divide a 

finite segment II' harmonically is a 

conic through I, I'. 

If P is any point on the locus, PI, PI' are, by Art. 166 (jS'), conjugate 

lines for the conic. Therefore I, I' are conjugate points, and the proposition 

is clearly equivalent to that of Art. 185. 

If the given conic is a parabola, the segment II' is a tangent, and the 
locus becomes the line II' and the line joining the points of contact of 
tangents from I and I'. 



24. The locus of the intersection 
of tangents to a parabola which 
meet at a given angle (not a right 
angle) is a hyperbola having double 
contact with the parabola. 



25. If a variable triangle is in- 
scribed in a circle, and two of its 
sides are parallel to given directions, 
the base envelops a concentric circle. 



26. If two circles are concentric, 
any chord of one which touches the 
other is bisected at the point of 
contact. 



27. If two circles touch one 
another, the straight line joining 
their centres passes through the 
point of contact. 
M. 



The locus of the intersection of 
tangents to a conic which divide a 
finite segment II' of a given tangent 
in a constant cross-ratio is a conic 
having double contact with the given 
conic at the points of contact of 
tangents from I and I', 

If a triangle is inscribed in a 
conic, having two of its sides passing 
through two fixed points on a given 
chord II', the base envelops a conic 
having double contact with the given 
conic at the points J, /'. Art. 276. 

If two conies have double contact, 
any chord of one which touches the 
other is divided harmonically by its 
point of contact, the chord of contact 
and the curve. Art. 266. 

If two conies touch one another, 
the line joining the poles of their 
common chord passes through their 
point of contact. 

18 



274 



CROSS-RATIO GEOMETRY 



[CH. XIX 



28. If two circles, centres t, t', 
touch one another at p, and a chord 
pqr is drawn, the radii tr, t'q are 
parallel ; and conversely. 



29. The locus of the centres of 
circles touching a given circle at a 
given point is a straight line passing 
through the given point. 



30. The centre of any circle 
which touches two intersecting 
straight lines lies on the bisector of 
the angle between them. 

31. Given the focus and two 
points on a conic, the directrix 
passes through one of two fixed 
points. 

32. The locus of the centres of 
circles which touch each of two 
parallel lines is a straight line 
parallel to the others and midway 
between them. 



33. If two circles intersect at a, 
h, and through a any double chord 
'paq is drawn, jog subtends a constant 
angle at h. 



Two conies touch one another at 
P, and T, T' are the poles of their 
common chord H', If any chord 
PQR is drawn through P, TR, T'Q 
will meet on II' ; and conversely, if 
TR, T'Q meet on II', QR passes 
through P. 

A system of conies is described 
touching a given conic at a given 
point P, and intersecting it in two 
fixed points I, I'. The locus of the 
poles of II' is a straight line passing 
through P. 

Given two tangents CA, CB and 
two points I, I' on a conic, the locus 
of the pole of the common chord II' 
is a double line ' of the involution 
pencil G {W, AB). Art. 288. 

Given two tangents and two 
points on a conic, their chord of 
contact will pass through one of two 
fixed points. Art. 288. 

I, I' are two fixed points, PQ, 
PR two fixed lines intersecting in P 
on II'. If a system of conies is 
described passing through I, I' and 
touching PQ, PR, the locus of 
the pole of II' is a straight line 
through P. 

Two conies intersect in the points 
I, I', A, B. Any chord through A 
meets the conies in P, Q. Then 
B {II'PQ) is constant. 

System of Co-axial circles. Pencil of conies. 

34. The centres of the circles The poles of any pair of common 

are collinear. chords are collinear. Art. 216. 

The poles lie along one of the sides of the common self -conjugate triangle 
P1P2P3 (Fig. 102), which may be called the line of poles corresponding to 
that pair. 



Exs. 28-42] 



EXAMPLES 



275 



Any pair of common chords 
forms a harmonic pencil with the 
two lines of poles through their 
vertex. Art. 214. 

Two given points are conjugate 
for only one conic of a pencil. Art. 
217. 

If two points are conjugate for 
two conies A, B, they are conjugate 
for the pencil to which A and B 
belong. Art. 218. 

A line of poles cuts the pencil of 
conies in a range in involution of 
which a pair of vertices of the self- 
conjugate triangle P1P2P3 are the 
double points. Art. 214. 

A system of conies which pass 
through three fixed points A, I, I', 
and have the poles of //' collinear, 
pass through a fourth point B. It L 
is the line of poles, cutting //' in K, 
then AB is the harmonic conjugate 
of AK for AI, AF, and B is the 
harmonic conjugate of A for the 
points where AB cuts L and H'. 

40. A system of circles which A system of conies which pass 
pass through a fixed point and have through three fixed points and have 
a pair of points conjugate are co- a pair of points conjugate form a 
axial. pencil. Art. 218. 

If the pair of conjugate points are i, i', the conies are all rectangular 
hyperbolas, and the fourth point through which they pass is the orthocentre 
of the triangle formed by the three given points. 

41. A system of circles which 
have two pairs of conjugate points is 
co-axial. 



35. The line of centres is per- 
pendicular to the radical axis. 



36. Two given points are con- 
jugate for only one circle of a co-axial 
system. 

37. If two points are conjugate 
for two circles, they are conjugate 
for the co-axial system to which they 
belong. 

38. The line of centres cuts the 
system in a range in involution of 
which the limiting point circles are 
the double points. 

39. A system of circles which 
pass through a fixed point a and have 
their centres collinear are co-axial, 
the second point h being such that ah 
is bisected at right angles by the line 
of centres. 



42. Three pairs of conjugate 
points determine a circle. 



A system of conies which 
through two fixed points and have 
two pairs of conjugate points form a 
pencil. Art. 218. 

Two points and three pairs of 
conjugates determine a conic. Art. 
218. 

18—2 



276 



CROSS-RATIO GEOMETRY 



[CH. XIX 



43. The polar s of any point on 
the radical axis intersect on the 
radical axis. 

44. The polar s of any point 
P will all pass through the same 
point P'. 

45. The radical axis bisects 

(1) The segment PP' in Ex. 44. 

(2) The common tangents of 
any pair of circles of the system. 

(3) The segment joining the 
limiting point circles L, L'. 

(4) The tangents from either 
limiting point circle to any circle of 
the system. 

46. If three circles are co-axial a 
common tangent to two of them is 
cut harmonically by the third. 



The polars of a point on any 
common chord intersect on that 
common chord. Art. 210. 

The polars of any point P will all 
pass through the same point P'. 
Art. 218. 

Any pair of common chords 
divide harmonically 

(1) The segment PP' in Ex. 44. 
Art. 218. 

(2) The common tangent of any 
pair of conies of the pencil. Art. 
213, Cor. 

(3) One of the sides of the com- 
mon self -con jugate triangle P1P2P3 . 
Art. 214. 

(4) The tangents from any vertex 
of the triangle P^P^P^ to any conic 
of the pencil. Art. 214. 

A common tangent to two conies 
of a pencil is cut harmonically by 
any other conic of the pencil. Art. 
213, Cor. 



47. The three radical axes of 
any three circles taken in pairs are 
concurrent in the radical centre. 



48. If in Ex. 47 c is the radical 
centre, and from c tangents are 
drawn to the three circles, the six 
points of contact will lie on a circle 
called the radical circle, whose centre 



If three conies have a common 
chord II', and the other common 
chord of the pair be drawn for each 
pair of conies, these latter common 
chords meet in a point which may 
be called the radical pole. Art. 215. 

If in Ex. 47 C is the radical pole, 
and from G tangents are drawn to the 
three conies, the six points of contact 
lie on a conic, which may be called 
the radical conic, which passes 
through the points J, Z', and has C 
for the pole ot II'. 



Exs. 43-50] 



EXAMPLES 



277 



Limiting point circles, L, TJ . 

49. (1) L, L' are inverse points 
for each circle. 

(2) The points of contact of a 
common tangent of two circles of the 
system subtend a right angle at L 
and L'. 

(3) If PQ is a common tangent 
to two of the circles, the circle on 
PQ as diameter passes through the 
points I/, L'. 



(4) If a transversal touches one 
circle at Q and cuts another at jR, *Sf, 
LQ bisects the angle UhS. 



(5) If the transversal in (4) 
passes through L, (LQRS) is har- 



Vertices of self-conjugate tri- 
angle P1P2P3. 

(1) See Ex. 38. 

(2) See Ex. 45 (3). 



(3) If QB is a common tangent 
of two conics» I, I' a pair of com- 
mon points, the conic described 
through Q, R, I, I', having QR 
and II' for conjugate lines passes 
through two vertices of the triangle 
P1P2P3. 

(4) A tangent to one conic at Q 
cuts another in R, S. If P is a 
vertex of the triangle P1P2P3 , PQ is 
a double ray of the involution pencil 
P{II', RS). 

(5) If the tangent in (4) passes 
through a vertex P of the triangle 
P1P2P3, (PQRS) is harmonic. 



Centres of similitude of two circles. 

50. (1) The centres of similitude 
and the centres of the two circles 
form a harmonic range. 

(2) The two circles and their 
circle of similitude are co-axial. 



(3) Any transversal through a 
centre of similitude meets the two 
circles in four points which lie by 
pairs at the extremities of parallel 
radii. 



Tangent vertices of two conies. 

(1) In Fig. 102, g^g^ is a com- 
mon chord, a, /3 its poles, Tj , Tg 
the corresponding tangent vertices, 
(a^T^T^) is harmonic. Art. 229. 

(2) The conic through Tj , 1\, 
g-^, g^ having T^T^ and g^g^^ for con- 
jugate lines passes through the 
points g^, g^ . 

(3) If in Fig. 106 a transversal 
through Tj meets the conic A in 
p, p' and the conic B in q, q\ and 
a', j3' are the poles of the common 
chord FF', a'p and §'q meet on FF\ 
&c. Art. 248. 



278 



CROSS-RATIO GEOMETRY 



[CH. XIX 



Orthogonal circles. 
51. A circle which is orthogonal 
to two circles has its centre on their 
radical axis. 



52. A circle which is orthogonal 
to two circles is orthogonal to the 
system which is co-axial with them. 

63. A system of circles which is 
orthogonal to two given circles is 
co-axial. 

54. If a system of conies has 
its centres coUinear and cuts a given 
circle orthogonally, it is co-axial. 



55. If a system of circles passes 
through a fixed point and cuts a 
given circle orthogonally, the system 
is co-axial. 

56. If c is any point on the 
radical axis of a co-axial system, and 
tangents are drawn from c to each 
circle, the points of contact will all 
lie on a circle which cuts the system 
orthogonally. 



57. Given a system of co-axial 
circles, there exists another co-axial 
system such that each circle of either 
system cuts every circle of the other 
system orthogonally. 



Contra-polar conies. 
A conic A which is contra-polar 
to two conies B, C passes through 
two of their common points, and the 
pole of the line joining them for the 
conic A lies on the line joining the 
other pair of common points of B 
and C. 

A conic which is contra-polar to 
two conies is contra-polar to the 
pencil to which they belong. 

A system of conies which is 
contra-polar to two given conies 
forms a pencil of conies. 

If a system of conies has two 
common points, and the poles of the 
line joining them are collinear, and 
also a conic through them is contra- 
polar to the system, the conies form 
a pencil. 

If a system of conies has three 
points common, and is contra-polar 
to a given conic through two of them, 
the system forms a pencil. 

If C is any point on one of a pair 
of common chords of a pencil of 
conies and tangents are drawn from 
G to each conic, the points of con- 
tact will all lie on a conic which 
passes through the two points where 
the other common chord of the pair 
intersects the conies, and is contra- 
polar to the pencil. 

Given a pencil of conies there 
exists another pencil such that each 
conic of either pencil is contra-polar 
to every conic of the other. The 
two pencils have two points /, I' 
common to both. 



Exs. 51-63] 



EXAMPLES 



279 



58. In Ex. 57 the line of centres 
of one system is the radical axis of 
the other. 



If QR, II' are & pair of common 
chords of the first pencil, and Q'R', 
II' a pair of the second, Q'R' is the 
locus of the poles of 11' for the first, 
and QR the locus of the poles of II' 
for the second. 



59. In Ex. 48, the radical circle 
of the three given circles cuts them 
orthogonally. 

60. If two circles cut one another 
orthogonally, any diameter of one is 
cut harmonically by the other. 



Given three conies which have a 
common chord, their radical conic is 
contra-polar to them. 

If two conies are contra-polar, 
any chord through the pole of one 
is cut harmonically by the other. 
Art. 205. 

Any conic through two of the 
common points of a pencil and two 
vertices of the common self -con jugate 
triangle is contra-polar to the pencil. 

Polax conic. 

Given a triangle ABC and two 

points 7, I', only one conic can be 

described passing through J, I' and 

having ABC self-conjugate. This 

may be called the polar conic of the 

triangle. 

To construct the conic, join AI meeting BG in D, and take E so that 

(AIDE) is harmonic. Then E is a point on the conic. Similarly by taking 

I with B and G we obtain two other points F, G on the curve. We thus 

obtain five points on the conic. 

63. The polar circle of a triangle The polar conic of a triangle 

divides the sides harmonically, and divides the sides harmonically, and 
is orthogonal to the circles on the is contra-polar to the three conies 
sides as diameters. constructed as follows. 

Given a triangle ABG and two points I, I'. Describe the conic passing 
through A, B, I, I' and having AB and II' for conjugate lines, and similarly 
for the groups B, G, J, I' and G, A, I, I'. These represent the circles on 
AB, BG, GA as diameters. Having two points common they have a radical 
pole which represents the orthocentre in the original figure, and the polar 
conic is contra-polar to them. 



61. Any circle through the 
limiting points of a co-axial system 
is orthogonal to the system, and 
conversely. 

Polar circle. 

62. Given a triangle ABG, only 
one circle can be described for which 
ABG is self-conjugate. This is called 
the polar circle of the triangle. 



280 



CROSS-RATIO GEOMETRY 



[CH. XI 



Confocal conies. 



64. Confocal conies cut at right 



Range of conies. (I, I' are now 
pair of opposite tangent vertices.)] 
In Fig. 102 (a^TiTa) is harmoni 
Art. 229. 



65. If from a point P on a conic 
tangents are drawn to a confocal, 
they make equal angles with the 
tangent to the given conic at P. 



Art. 228. 



66. The locus of the pole of a 
straight line L for a system of con- 
f ocals is a straight line perpendicular 
toL. 



Art. 233. 



APPENDIX II 

pascal's theorem proved for the conic and line-pair 
by the methods of euclid and apollonius 



In Figs. 128, 129, a, 6, c and a\ b\ c' are any six points on the 
conic, or by threes on the pair of lines L, L' . 

a, y8, y are the intersections of (5c', h'c), [ca', ca), {ab', a'h). 
It is required to prove that a, ^, y are collinear. 




Fig. 128. 



282 



CROSS-RATIO GEOMETRY 




Fig. 129. 




APPENDIX II 



283 



Through a draw ae parallel to ca'. In the conic, is the 
centre, hst, b's't', and OA are parallel to ca', and OB is parallel 
to aa'. Complete the figures by joining points as required. 



I. For the conic. 
By similar triangles 



eh :b' 



Again 



and 



he : cb' 
= am : mb' 
= aa' : a't', 
a'm : b't' = aa' : a^', 
i . a'm : aa'^ = b's' . b't' : at' . a!t' 

^OA^'. OW 

ck : sb — ec : es 

= aa' : at, 
ck : aa' = sb : at, 
al : aa' = bt : ta', 
ck . al : aa'^ = 8b .bt : at. ta' 

= 0A^ : OB^ 

ck.al = eh. a'm. 



.(1). 



(2); 



by (1) and (2) 

II. For the line-pair. 

By similar triangles 

ck \ ae = kb : be 

= ka' : le, 

ck : ka' = ae : le, 

ck : ca' = ae : al, 

ck.al = ca'.ae 

Again, from the similar triangles b'eh, b'a'c, 
eh : a'c = b'e : b'a' 
— ae : a'm, 
.*. eh . a'm = a'c .ae-ck. al, by (3), 



,(3). 



284 



CROSS-RATIO GEOMETRY 



.*. in both the conic and the line-pair, 

eh : ck = al : a'm, 
:. eg -.gk^ly: ya\ 

ek : gk - la : ya' (4), 

and bk : be = ha! : hi, 

hk : ek = ha' : a'l (5), 

.'. by (4) and (5), hk : gk = ha' : ya', 

.'. 7^ is parallel to a'c or ae. 

Now y is the intersection of (ah', ah), and g of (eh, h'c). 
Therefore if we introduce the point c' in the place of h', and 
if n be the intersection of (ac', ah), and n that of {eh, cc), nn 
will be parallel to a'c or ae, and therefore also to yg. 

ftp : pc = nr : rn' 

= n • m 

:. Pp'.yq=pc:qg 
- ap : aq, 
.-. ftp : pa - yq : qa, 
.*. a, ft, y are collinear. 



INDEX 



The numbers refer to pages 



Aldrich 75 

Anharmonic function or ratio 2 
, , properties of points and 

tangents 139 
Apollonius viii, ix, 51, 74, 115, 136, 

138, 139, 144, 146, 147, 165 
Aristaeus x 
Axis of a range 2 

„ perspective 16, 20 

Ball, Short History of Mathematics 

147 
Barrow viii 
Base of a range 2 
Bennett 53 
Bernard 75 
Bomiscus 125 
Borellus ix 

Brianchon vii, 40, 158, 161 
Burrow ix 

Carnot, Geometrie de position 1 
Centre of involution 89 

,, „ at infinity 96 

„ pencil 2 

,, perspective 15, 20 

Characteristic of a range 29 

„ „ an involution 93 

Chasles 6, 76, 82, 120, 139 

,, Apergu Historique v, 2, 89, 
130, 140, 159 
Chasles, Geovi. Sup. v, 2, 69, 72, 

74, 86, 130, 140 
Chasles, Porismes cfEiuclide 122 
,, Traite des Sect. Con. v. 



vii, 2, 130, 140, 181, 184, 201, 

206, 209, 246 
Chord vertex 203 
Circular points 100, 176, 180, Chap. 

XIX 
Classes of porisms 118 
Clifford 2 

Co-axial circles Chap, XIX 
,, ranges Chap. VII 
triangles 19, 37, 38 
Common chords of two conies Chap. 

XIV 
Common point of two ranges 17, 31 
„ points, Chasles' method of 

finding 69, 72 
Common points, Lodge's method 70 
,, ,, cross-ratio (aa'ef) 

constant 62, 163 
Common points, discrimination of 

64 
Common points of co-axial ranges 

Chap. VII 
Common points of conic-ranges 161, 

181 
Common points of similar ranges 

63 
Common ray of two pencils 18, 33 
,, rays of concentric pencils 

65 
Conic-pencil 131 

,, in involution 168 

Conic-range 161 

Conies inscribed in quadrilateral 210 
Conic through five points 135, 143, 

255 



286 



INDEX 



I 



Conic touching five lines 136, 144, 

257 
Conjugate lines and points 169-176, 

197, 198 
Conjugate lines for range of conies 

213 
Conjugate lines through two points 

179 
Conjugate points 89 

,, ,, for pencil of conies 

207 
Constant of correspondence 48 
,, „ homology 222, 230 
Construction of centre of involution 

92 
Construction of common chords 

Chap. XIV 
Construction of conies satisfying 

conditions Chap. XVIII 
Construction of fourth harmonic of 

three points 22, 25 
Construction of fourth harmonic of 

three rays 23, 108 
Construction of fourth point of 

range 7, 29 
Construction of fourth ray of pencil 

11, 34 
Construction of I and J' 30, 41, 42 
„ ,, involution range on 

line 93, 98 
Construction of involution range on 

conic 169 
Construction of range homographic 

to co-axial range 67, 71, 164 
Construction of range homographic 

to conic-range 162 
Construction of range homographic 

to given range 29 
Construction of range in perspective 

with two others 33 
Construction of ray corresponding 

to 00' 43 
Construction of self- conjugate tri- 
angle 173 
Contra-polar conies 193, Chap. XIX 
Co-polar and co-axial triangles 19, 

37, 38 
Corresponding chords and tangent 

vertices 212 
Cremona 84, 86 
Cross-axis and cross-centre Chap. V 



Cross-axis parallel to IJ' 42 
,, property 39 

Cross- centre property 39 

Cross-ratio and three points given 7 
,, cannot equal 1, or oo 6 

,, of conic-pencil 132 

,, ,, four conies 209, 215 

,, ,, ,, lines 10, 11 

,, ,, ,, points 2 

,, ,, ,, tangents 133 

,, ,, pencil 10 

,, reciprocal of 3 

,, when unaltered 3 

Cross-ratios, four positive, two 
negative 5 

Cross-ratios, represented geometri- 
cally 9 

Data (in generalised projection) 

263-266 
Delbalat 140 
Desargues 149 

„ Theorem vii, 182 
„ ,, correlative 182 

Descartes 147 

Determinate Section ix, 75 
Director circle, directrix 180 
Double contact conies Chap. XVII 
,, points 89 

„ ,, discrimination of 98 

Duality viii 
Durell, Plane Geometi-y, Part ii, x 

Ecchellensis ix 
{efO'J') is harmonic 61 
Eleven-point conic 210 

„ tangent conic 216 
Equicross ranges and pencils Chap. 

II 
Eratosthenes x 
Euclid's Data ix 

,, Porisms viii, ix, 26, 36, 40, 

Ap. I 
Euler's Theorem 4 

Factorising homography 53 

Fermat ix, 121 

Filon 40 

Focus of a conic 178 

Fregier's Theorem 178 




INDEX 



287 



Geometrical representation of cross- 
ratios 9 
Ghetaldus ix 
Gregory 75 

Halley ix, 51, 75, 121 
Harmonic conjugate 21 
,, homology 231 

,, properties of quadrangle 

and quadrilateral 106 
Harmonic range determines har- 
monic pencil 24 
Harmonic ratio Chap. HI 

,, section and involution 
103 
Harmotomic conies 195 
Heawood x 
Hesse 201 
nomographic co-axial ranges Chap. 

VII 
nomographic equations 48 
,, ranges 28, 48 

,, ,, and pencils 

Chap. IV 
Homologue of a conic 223 

,, ,, straight line 222 

Homology Chap. XV 
Home vi, 130 
Horsley ix 

Identical ranges 15, 32 

I, J', construction for 30, 41, 42 

IJ' parallel to cross-axis 42 

Inclinations ix 

Inscribe a triangle in a given tri- 
angle 81 

Intersection of two chords 168 

,, ,, tangents dividing a 

segment harmonically 180 

Involution Chaps. IX, X 
,, equations 94 

,, of conic-pencil 168 

,, on a tangent 175 

,, pencil 99 

„ range 87 

Isotropic lines 101 

Lahire 165, 167, 184 
Lame 207 
Lawson ix, 76 
Local Theorem 116 



Locus ad quatuor lineas 146 

, , of centres of pencil of conies 
210 

Lodge X, 69, 131, 196, 203 

Lodge's circles 71, 98 

,, method of finding common 
points vi, 69 

Lodge's method of finding corre- 
sponding points 71 

Lodge's method of finding double 
and conjugate points 98 

Maclaurin 159, 184 

Maclaurin's Theorem 159, 160 

,, ,, correlative 161 

Mathematical Gazette x, 69, 147, 
217 

Mathematical Society {Lond.) Proc. 
53 

Menelaus 13 

Theorem 12, 40 

Metrical properties of homographic 
ranges Chap. VI 

Mid-points of ef and IJ' coincide 61 

Milne and Davis, Geom, Con, 139, 
148 

Milne, Dr W. P., Theorem 217 
,, „ ,, ,, correla- 
tive 217 

Mobius V, 2 

Newton's Description of a conic 159 
Principia 59, 139, 144, 
147, 159 
Newton's Universal Arithmetic 144 

One-to-one correspondence 53, 98 
Orthogonal circles Chap. XIX 
,, pencil 100 

Pappus viii, 15, 25, 26, 36, 40, 51, 

74, 76, 95, 109, 114, 115, 121, 

122, 144, 147, 156 
Pascal 40, 156 

„ line 40, 163, 164, 187, 188 

,, Theorem vii, 126, 128, 149, 

156, 160, 161 
Pencil 2 

,, of conies 206 
Pencils homographic to same pencil 

32 



288 



INDEX 



Pencils in perspective 17, 33 

,, superposable or identical 32 

Perspective Chap. II 

Pole and polar 110, 167 

Poncelet 120 

„ Prop. Proj. 184, 208, 209, 
210, 246, 249, 250 

Porisms of Euclid ix, Ap. I 

Potts 123 

Power of correspondence 48 
,, involution 93 

Projection viii, 141-143, Chap. XIX 

Proportional Section ix, 51, 80 

Quadrangle and quadrilateral 105, 

184 
Quadrangle cut by a transversal 109 
,, inscribed in a conic 

192 
Quadrilateral circumscribing a conic 

191 

Bange 2 

,, of conies 213 

, , , , poles and pencil of polars 

homographic 171 
Ranges homographic to same range 

29 
Ranges in perspective 15, 31 
Reciprocal conic for a pencil 209 

,, of cross-ratio 3 

Relations between common chords 

and tangent vertices 226 
Relations between four constants of 

homology 230 
Relations between the segments of 

harmonic range 25 
Relations between the six cross- 
ratios 5 
Rotation angle and centres 67 
Rule of false position 82 
,, involution 94 



Russell, Treatise on Pure Geometry 
X, 231 



Salmon's Conies 38, 246, 250 

Schooten ix 

Segments of harmonic range 

lations between 25 
Self-conjugate triangle 169, 189, 

190, 191, 205 
Self-conjugate triangle, construction 

of 173 
Sign first employed in geometry 1 
Similar ranges 15, 51 

,, ,, common points 63 

Simson ix, 76, 117, 121, 122 
Snell ix, 76 
Spatial Section ix, 78 
Steiner 48 
Sturm 206 



T {aa'ef) constant 187 

Tangencies ix 

Tangent vertices 211 

Taylor 210 

Three conies having common chord 

205 
Three conies having common tangent 

vertex 212 
Townsend 86, 252 
Triangles co-axial and co-polar 19, 

37, 38 
Triangles circumscribing a conic 189 
,, inscribed in a conic 188 
,, in perspective 19 
,, self -con jugate 169 
,, ,, construction 

of 173 

ijTTTlOU irapvTTTioif 121 

Vertex of pencil 2 
Vieta ix 






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